Average Error: 31.2 → 0.6
Time: 18.2s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right) \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right)\right)}\]

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 31.2

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log 10}}\]
  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 pow10.6

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

  6. Applied log-pow0.6

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

  7. Applied times-frac0.5

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

  9. Applied div-inv0.4

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

  11. Applied add-cbrt-cube0.6

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\sqrt[3]{\left(\left(\log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\sqrt{\log 10}}\right) \cdot \left(\log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\sqrt{\log 10}}\right)\right) \cdot \left(\log \left(\sqrt{re^2 + im^2}^*\right) \cdot \frac{1}{\sqrt{\log 10}}\right)}}\]

  12. Final simplification0.6

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \sqrt[3]{\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right) \cdot \left(\left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right) \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt{re^2 + im^2}^*\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019022 +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: 17.1s)Debug log

sample44.0ms

Algorithm
intervals
Results
32.0ms256×body80valid

simplify13.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
13.0ms
(/ (log (sqrt (+ (* re re) (* im im)))) (log 10))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.6b

localize23.0ms

Local error

Found 2 expressions with local error:

0.6b
(/ (log (hypot re im)) (log 10))
0.0b
(hypot re im)

rewrite9.0ms

Algorithm
rewrite-expression-head
Rules
12×times-frac
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
6×pow1
4×associate-/l*
4×log-pow
4×add-exp-log
4×add-cbrt-cube
3×associate-/r*
2×add-log-exp
2×log1p-expm1-u
2×expm1-log1p-u
1×div-inv
1×div-exp
1×frac-2neg
1×clear-num
1×hypot-udef
1×cbrt-undiv
Counts
2 → 43
Calls
2 calls:
Slowest
8.0ms
(/ (log (hypot re im)) (log 10))
0.0ms
(hypot re im)

series232.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
200.0ms
(/ (log (hypot re im)) (log 10))
33.0ms
(hypot re im)

simplify493.0ms

Counts
37 → 49
Calls
37 calls:
Slowest
135.0ms
(* -1 (/ (log (/ -1 re)) (log 10)))
113.0ms
(/ (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (log 10) (log 10)) (log 10)))
104.0ms
(* -1 (/ (log (/ 1 re)) (log 10)))
37.0ms
(/ (* (cbrt (log (hypot re im))) (cbrt (log (hypot re im)))) (* (cbrt (log 10)) (cbrt (log 10))))
18.0ms
(/ (* (cbrt (log (hypot re im))) (cbrt (log (hypot re im)))) 1)

prune495.0ms

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 0.1b

localize12.0ms

Local error

Found 3 expressions with local error:

0.5b
(/ (log (hypot re im)) (sqrt (log 10)))
0.3b
(* (/ 1 (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10))))
0.0b
(hypot re im)

rewrite14.0ms

Algorithm
rewrite-expression-head
Rules
48×times-frac
36×add-cube-cbrt
36×*-un-lft-identity
36×add-sqr-sqrt
28×associate-*r*
27×sqrt-prod
18×pow1
15×add-exp-log
13×log-pow
10×add-cbrt-cube
6×associate-/r*
4×associate-/l*
4×prod-exp
4×associate-*l*
3×add-log-exp
3×div-inv
3×log1p-expm1-u
3×div-exp
3×expm1-log1p-u
2×cbrt-unprod
2×rec-exp
2×cbrt-undiv
1×*-commutative
1×associate-*r/
1×associate-*l/
1×pow-prod-down
1×frac-2neg
1×clear-num
1×hypot-udef
1×frac-times
Counts
3 → 110
Calls
3 calls:
Slowest
7.0ms
(* (/ 1 (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10))))
6.0ms
(/ (log (hypot re im)) (sqrt (log 10)))
0.0ms
(hypot re im)

series390.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
191.0ms
(/ (log (hypot re im)) (sqrt (log 10)))
159.0ms
(* (/ 1 (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10))))
40.0ms
(hypot re im)

simplify1.8s

Counts
108 → 119
Calls
108 calls:
Slowest
421.0ms
(* (* (* (/ 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)))))
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)))))
124.0ms
(* -1 (/ (log (/ -1 re)) (log 10)))
119.0ms
(/ (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (sqrt (log 10)) (sqrt (log 10))) (sqrt (log 10))))
103.0ms
(+ (log (/ 1 (sqrt (log 10)))) (- (log (log (hypot re im))) (log (sqrt (log 10)))))

prune1.4s

Pruning

14 alts after pruning (14 fresh and 0 done)

Merged error: 0.0b

localize5.0ms

Local error

Found 3 expressions with local error:

0.3b
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
0.3b
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
0.0b
(hypot re im)

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
23×add-exp-log
12×prod-exp
11×pow1
10×add-cbrt-cube
8×associate-*l*
6×add-cube-cbrt
6×*-un-lft-identity
6×rec-exp
6×add-sqr-sqrt
5×associate-*r/
5×associate-*r*
4×cbrt-unprod
4×pow-prod-down
3×add-log-exp
3×log1p-expm1-u
3×un-div-inv
3×expm1-log1p-u
2×div-inv
2×*-commutative
2×frac-times
1×log-pow
1×associate-*l/
1×hypot-udef
Counts
3 → 64
Calls
3 calls:
Slowest
11.0ms
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
5.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
0.0ms
(hypot re im)

series364.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
197.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
143.0ms
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
23.0ms
(hypot re im)

simplify2.3s

Counts
41 → 73
Calls
41 calls:
Slowest
456.0ms
(* (* (* (/ 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))))))
357.0ms
(* (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))))
297.0ms
(* (* (* (/ 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))))))
118.0ms
(+ (log (/ 1 (sqrt (log 10)))) (log (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
116.0ms
(* -1 (* (log (/ -1 re)) (sqrt (/ 1 (log 10)))))

prune898.0ms

Pruning

12 alts after pruning (11 fresh and 1 done)

Merged error: 0b

localize8.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sqrt (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
0.3b
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
0.3b
(* (sqrt (/ 1 (sqrt (log 10)))) (* (sqrt (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
0.0b
(hypot re im)

rewrite61.0ms

Algorithm
rewrite-expression-head
Rules
29×add-exp-log
21×pow1
20×add-cbrt-cube
18×associate-*r/
18×associate-*l*
16×prod-exp
15×sqrt-div
11×frac-times
10×cbrt-unprod
10×add-cube-cbrt
10×pow-prod-down
10×*-un-lft-identity
10×add-sqr-sqrt
8×sqrt-prod
7×un-div-inv
6×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×associate-*l/
4×expm1-log1p-u
3×div-inv
3×*-commutative
3×rec-exp
1×log-pow
1×hypot-udef
Counts
4 → 103
Calls
4 calls:
Slowest
31.0ms
(* (sqrt (/ 1 (sqrt (log 10)))) (* (sqrt (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
19.0ms
(* (sqrt (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
9.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
0.0ms
(hypot re im)

series909.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
546.0ms
(* (sqrt (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
174.0ms
(* (sqrt (/ 1 (sqrt (log 10)))) (* (sqrt (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
166.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
23.0ms
(hypot re im)

simplify4.8s

Counts
78 → 115
Calls
78 calls:
Slowest
926.0ms
(* (* (* (sqrt (/ 1 (sqrt (log 10)))) (sqrt (/ 1 (sqrt (log 10))))) (sqrt (/ 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))))))
339.0ms
(+ (log (sqrt (/ 1 (sqrt (log 10))))) (+ (log (sqrt (/ 1 (sqrt (log 10))))) (log (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))))
279.0ms
(* (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))))
236.0ms
(+ (log (sqrt (/ 1 (sqrt (log 10))))) (+ (log (sqrt (/ 1 (sqrt (log 10))))) (+ (log (log (hypot re im))) (log (/ 1 (sqrt (log 10)))))))
222.0ms
(+ (log (sqrt (/ 1 (sqrt (log 10))))) (log (* (sqrt (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))))

prune1.3s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

regimes154.0ms

Accuracy

0% (0.6b remaining)

Error of 0.6b against oracle of 0.0b and baseline of 0.6b

bsearch4.0ms

end0.0ms

sample1.3s

Algorithm
intervals
Results
1.2s8000×body80valid