Average Error: 31.2 → 0.4
Time: 49.8s
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 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 *-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-frac0.6

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\sqrt{\log 10}}}\]
  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 2019016 +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: 47.8s)Debug log

sample48.0ms

Algorithm
intervals
Results
256×(pre true 80)
256×(body real 80)

simplify27.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
26.0ms
(/ (log (sqrt (+ (* re re) (* im im)))) (log 10))

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.6b

localize27.0ms

Local error

Found 2 expressions with local error:

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

rewrite15.0ms

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
13.0ms
(/ (log (hypot re im)) (log 10))
0.0ms
(hypot re im)

series192.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
152.0ms
(/ (log (hypot re im)) (log 10))
39.0ms
(hypot re im)

simplify559.0ms

Counts
37 → 49
Calls
37 calls:
Slowest
176.0ms
(/ (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (log 10) (log 10)) (log 10)))
110.0ms
(* -1 (/ (log (/ 1 re)) (log 10)))
81.0ms
(* -1 (/ (log (/ -1 re)) (log 10)))
36.0ms
(/ (* (cbrt (log (hypot re im))) (cbrt (log (hypot re im)))) (* (cbrt (log 10)) (cbrt (log 10))))
22.0ms
(/ (* (cbrt (log (hypot re im))) (cbrt (log (hypot re im)))) 1)

prune567.0ms

Pruning

14 alts after pruning (13 fresh and 1 done)

Merged error: 0.1b

localize10.0ms

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))))
5.0ms
(/ (log (hypot re im)) (sqrt (log 10)))
0.0ms
(hypot re im)

series427.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
224.0ms
(/ (log (hypot re im)) (sqrt (log 10)))
178.0ms
(* (/ 1 (sqrt (log 10))) (/ (log (hypot re im)) (sqrt (log 10))))
25.0ms
(hypot re im)

simplify2.1s

Counts
108 → 119
Calls
108 calls:
Slowest
472.0ms
(* (* (* (/ 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)))))
141.0ms
(/ (* (* (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 (/ 1 (log 10))) (log (/ 1 re))))
115.0ms
(+ (log (/ 1 (sqrt (log 10)))) (log (/ (log (hypot re im)) (sqrt (log 10)))))
113.0ms
(+ (log (/ 1 (sqrt (log 10)))) (- (log (log (hypot re im))) (log (sqrt (log 10)))))

prune1.4s

Pruning

15 alts after pruning (14 fresh and 1 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 3 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))))
0.0b
(hypot re im)

rewrite19.0ms

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
12.0ms
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
7.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
0.0ms
(hypot re im)

series430.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
227.0ms
(* (log (hypot re im)) (/ 1 (sqrt (log 10))))
172.0ms
(* (/ 1 (sqrt (log 10))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
31.0ms
(hypot re im)

simplify2.3s

Counts
41 → 73
Calls
41 calls:
Slowest
471.0ms
(* (* (* (/ 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))))))
323.0ms
(* (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10)))))
254.0ms
(* (* (* (/ 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))))))
148.0ms
(+ (log (/ 1 (sqrt (log 10)))) (log (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
118.0ms
(* -1 (/ (log (/ -1 re)) (log 10)))

prune950.0ms

Pruning

15 alts after pruning (13 fresh and 2 done)

Merged error: 0.0b

localize24.0ms

Local error

Found 4 expressions with local error:

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

rewrite276.0ms

Algorithm
rewrite-expression-head
Rules
181×add-exp-log
174×associate-*r/
142×frac-times
140×pow1
136×prod-exp
114×un-div-inv
78×cbrt-div
67×pow-prod-down
59×associate-*l/
58×add-cbrt-cube
40×cbrt-unprod
35×rec-exp
30×pow-prod-up
15×pow-plus
6×add-cube-cbrt
6×associate-*l*
6×associate-*r*
6×*-un-lft-identity
6×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
4×pow2
3×*-commutative
2×pow3
2×cbrt-prod
1×div-inv
1×rem-cbrt-cube
1×pow1/3
Counts
4 → 267
Calls
4 calls:
Slowest
97.0ms
(* (/ 1 (sqrt (log 10))) (cbrt (* (* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))))
73.0ms
(* (* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
58.0ms
(cbrt (* (* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
36.0ms
(* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))

series915.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
379.0ms
(* (* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
207.0ms
(* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))
206.0ms
(cbrt (* (* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
122.0ms
(* (/ 1 (sqrt (log 10))) (cbrt (* (* (* (log (hypot re im)) (/ 1 (sqrt (log 10)))) (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (* (log (hypot re im)) (/ 1 (sqrt (log 10)))))))

simplify31.8s

Counts
300 → 279
Calls
300 calls:
Slowest
801.0ms
(* (* (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10))))) (* (* (* (log (hypot re im)) (log (hypot re im))) (log (hypot re im))) (* (* (/ 1 (sqrt (log 10))) (/ 1 (sqrt (log 10)))) (/ 1 (sqrt (log 10))))))
552.0ms
(+ (+ (log (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (+ (log (log (hypot re im))) (log (/ 1 (sqrt (log 10)))))) (log (* (log (hypot re im)) (/ 1 (sqrt (log 10))))))
520.0ms
(+ (+ (+ (log (log (hypot re im))) (- (log (sqrt (log 10))))) (+ (log (log (hypot re im))) (- (log (sqrt (log 10)))))) (+ (log (log (hypot re im))) (log (/ 1 (sqrt (log 10))))))
479.0ms
(+ (+ (+ (log (log (hypot re im))) (- (log (sqrt (log 10))))) (+ (log (log (hypot re im))) (log (/ 1 (sqrt (log 10)))))) (+ (log (log (hypot re im))) (log (/ 1 (sqrt (log 10))))))
478.0ms
(+ (+ (log (* (log (hypot re im)) (/ 1 (sqrt (log 10))))) (+ (log (log (hypot re im))) (log (/ 1 (sqrt (log 10)))))) (+ (log (log (hypot re im))) (- (log (sqrt (log 10))))))

prune3.7s

Pruning

13 alts after pruning (11 fresh and 2 done)

Merged error: 0b

regimes194.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch2.0ms

end0.0ms

sample1.8s

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)