Average Error: 30.7 → 17.4
Time: 52.0s
Precision: 64
Internal Precision: 128
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.153348860420522 \cdot 10^{+45}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -1.2644302017014641 \cdot 10^{-201} \lor \neg \left(re \le -4.182444839958769 \cdot 10^{-214}\right):\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base}{\log base \cdot \log base}\\ \end{array}\]

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. Split input into 3 regimes

  2. if re < -2.153348860420522e+45

    1. Initial program 43.0

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

    2. Simplified43.0

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base}}\]

    3. Taylor expanded around -inf 62.8

      \[\leadsto \color{blue}{-1 \cdot \frac{\log \left(\frac{-1}{re}\right)}{\log -1 - \log \left(\frac{-1}{base}\right)}}\]

    4. Simplified11.1

      \[\leadsto \color{blue}{\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)}\]

    5. Taylor expanded around -inf 62.8

      \[\leadsto \color{blue}{\frac{-1}{\log -1 - \log \left(\frac{-1}{base}\right)}} \cdot \log \left(\frac{-1}{re}\right)\]

    6. Simplified11.1

      \[\leadsto \color{blue}{\frac{-1}{\log base}} \cdot \log \left(\frac{-1}{re}\right)\]

    if -2.153348860420522e+45 < re < -1.2644302017014641e-201 or -4.182444839958769e-214 < re

    1. Initial program 21.3

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

    2. Simplified21.3

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

    4. Applied times-frac21.2

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log base} \cdot \frac{\log base}{\log base}}\]

    5. Simplified21.2

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

    if -1.2644302017014641e-201 < re < -4.182444839958769e-214

    1. Initial program 30.5

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

    2. Simplified30.5

      \[\leadsto \color{blue}{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base}{\log base \cdot \log base}}\]

    3. Taylor expanded around -inf 47.3

      \[\leadsto \frac{\log \color{blue}{\left(-1 \cdot re\right)} \cdot \log base}{\log base \cdot \log base}\]

    4. Simplified47.3

      \[\leadsto \frac{\log \color{blue}{\left(-re\right)} \cdot \log base}{\log base \cdot \log base}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.153348860420522 \cdot 10^{+45}:\\ \;\;\;\;\frac{-1}{\log base} \cdot \log \left(\frac{-1}{re}\right)\\ \mathbf{elif}\;re \le -1.2644302017014641 \cdot 10^{-201} \lor \neg \left(re \le -4.182444839958769 \cdot 10^{-214}\right):\\ \;\;\;\;\frac{\log \left(\sqrt{im \cdot im + re \cdot re}\right)}{\log base}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(-re\right) \cdot \log base}{\log base \cdot \log base}\\ \end{array}\]

Reproduce

herbie shell --seed 2019018 
(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: 49.4s)Debug log

sample1.1s

Algorithm
intervals
Results
743.0ms252×body10240exit
160.0ms492×body80nan
88.0ms257×body80valid

simplify6.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
6.0ms
(/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0)) (+ (* (log base) (log base)) (* 0 0)))

prune11.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 30.2b

localize50.0ms

Local error

Found 4 expressions with local error:

28.7b
(sqrt (+ (* re re) (* im im)))
0.5b
(* (log base) (log base))
0.4b
(/ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (log base) (log base)))
0.4b
(* (log (sqrt (+ (* re re) (* im im)))) (log base))

rewrite41.0ms

Algorithm
rewrite-expression-head
Rules
20×add-exp-log
20×add-cbrt-cube
12×pow1
10×add-sqr-sqrt
9×add-cube-cbrt
9×*-un-lft-identity
8×associate-*l*
6×cbrt-unprod
6×prod-exp
6×associate-*r*
4×add-log-exp
4×div-exp
4×cbrt-undiv
3×sqrt-prod
2×*-commutative
2×log-pow
2×pow-prod-down
2×sqrt-div
2×pow1/2
1×associate-/l*
1×div-inv
1×flip-+
1×pow-prod-up
1×pow-plus
1×times-frac
1×flip3-+
1×frac-2neg
1×associate-/r*
1×clear-num
1×rem-sqrt-square
1×pow2
Counts
4 → 74
Calls
4 calls:
Slowest
19.0ms
(/ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (log base) (log base)))
10.0ms
(* (log (sqrt (+ (* re re) (* im im)))) (log base))
7.0ms
(* (log base) (log base))
4.0ms
(sqrt (+ (* re re) (* im im)))

series416.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
176.0ms
(* (log base) (log base))
121.0ms
(/ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (log base) (log base)))
95.0ms
(* (log (sqrt (+ (* re re) (* im im)))) (log base))
24.0ms
(sqrt (+ (* re re) (* im im)))

simplify3.1s

Counts
49 → 86
Calls
49 calls:
Slowest
446.0ms
(- (+ (log (log (sqrt (+ (* re re) (* im im))))) (log (log base))) (+ (log (log base)) (log (log base))))
391.0ms
(- (+ (log (log (sqrt (+ (* re re) (* im im))))) (log (log base))) (log (* (log base) (log base))))
339.0ms
(pow (- (log -1) (log (/ -1 base))) 2)
308.0ms
(* -1 (* (log (/ -1 re)) (- (log -1) (log (/ -1 base)))))
227.0ms
(- (log (* (log (sqrt (+ (* re re) (* im im)))) (log base))) (log (* (log base) (log base))))

prune1.1s

Pruning

12 alts after pruning (12 fresh and 0 done)

Merged error: 5.7b

localize24.0ms

Local error

Found 2 expressions with local error:

0.3b
(* (/ -1 (log base)) (log (/ -1 re)))
0.3b
(/ -1 (log base))

rewrite10.0ms

Algorithm
rewrite-expression-head
Rules
8×log-prod
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow1
4×div-inv
4×add-exp-log
4×associate-*l*
4×associate-*r*
4×add-cbrt-cube
4×distribute-lft-in
4×distribute-rgt-in
3×associate-/r*
2×add-log-exp
1×cbrt-unprod
1×*-commutative
1×log-pow
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×frac-2neg
1×clear-num
Counts
2 → 41
Calls
2 calls:
Slowest
8.0ms
(* (/ -1 (log base)) (log (/ -1 re)))
1.0ms
(/ -1 (log base))

series326.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
217.0ms
(/ -1 (log base))
109.0ms
(* (/ -1 (log base)) (log (/ -1 re)))

simplify1.1s

Counts
27 → 47
Calls
27 calls:
Slowest
450.0ms
(* (* (* (/ -1 (log base)) (/ -1 (log base))) (/ -1 (log base))) (* (* (log (/ -1 re)) (log (/ -1 re))) (log (/ -1 re))))
150.0ms
(* -1 (/ (- (log -1) (log re)) (log base)))
140.0ms
(* -1 (/ (log (/ -1 re)) (- (log -1) (log (/ -1 base)))))
54.0ms
(* (log (/ 1 re)) (/ -1 (log base)))
51.0ms
(/ (+ (log (/ 1 re)) (log -1)) (log (/ 1 base)))

prune511.0ms

Pruning

21 alts after pruning (21 fresh and 0 done)

Merged error: 5.6b

localize1.0ms

Local error

Found 2 expressions with local error:

0.3b
(* (/ -1 (log base)) (log (/ -1 re)))
0.3b
(/ -1 (log base))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
8×log-prod
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
5×pow1
4×div-inv
4×add-exp-log
4×associate-*l*
4×associate-*r*
4×add-cbrt-cube
4×distribute-lft-in
4×distribute-rgt-in
3×associate-/r*
2×add-log-exp
1×cbrt-unprod
1×*-commutative
1×log-pow
1×prod-exp
1×associate-*l/
1×pow-prod-down
1×frac-2neg
1×clear-num
Counts
2 → 41
Calls
2 calls:
Slowest
5.0ms
(* (/ -1 (log base)) (log (/ -1 re)))
0.0ms
(/ -1 (log base))

series256.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
164.0ms
(/ -1 (log base))
92.0ms
(* (/ -1 (log base)) (log (/ -1 re)))

simplify1.1s

Counts
27 → 47
Calls
27 calls:
Slowest
400.0ms
(* (* (* (/ -1 (log base)) (/ -1 (log base))) (/ -1 (log base))) (* (* (log (/ -1 re)) (log (/ -1 re))) (log (/ -1 re))))
143.0ms
(* -1 (/ (log (/ -1 re)) (- (log -1) (log (/ -1 base)))))
128.0ms
(* -1 (/ (- (log -1) (log re)) (log base)))
60.0ms
(* (log (/ 1 re)) (/ -1 (log base)))
51.0ms
(/ (+ (log (/ 1 re)) (log -1)) (log (/ 1 base)))

prune552.0ms

Pruning

21 alts after pruning (20 fresh and 1 done)

Merged error: 5.6b

localize41.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (/ -1 re))
0.6b
(cbrt (/ -1 re))
0.6b
(cbrt (/ -1 re))
0.5b
(* (cbrt (/ -1 re)) (cbrt (/ -1 re)))

rewrite12.0ms

Algorithm
rewrite-expression-head
Rules
20×cbrt-prod
11×add-cube-cbrt
11×*-un-lft-identity
11×add-sqr-sqrt
9×pow1
7×associate-*l*
7×associate-*r*
7×pow1/3
7×cbrt-div
6×add-exp-log
5×div-inv
4×add-log-exp
4×add-cbrt-cube
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×*-commutative
1×associate-*r/
1×prod-exp
1×associate-*l/
1×pow-plus
1×frac-times
1×pow2
Counts
4 → 72
Calls
4 calls:
Slowest
8.0ms
(* (cbrt (/ -1 re)) (cbrt (/ -1 re)))
1.0ms
(cbrt (/ -1 re))
1.0ms
(cbrt (/ -1 re))
1.0ms
(cbrt (/ -1 re))

series1.3s

Counts
4 → 12
Calls
4 calls:
Slowest
385.0ms
(cbrt (/ -1 re))
368.0ms
(* (cbrt (/ -1 re)) (cbrt (/ -1 re)))
279.0ms
(cbrt (/ -1 re))
273.0ms
(cbrt (/ -1 re))

simplify204.0ms

Counts
53 → 84
Calls
53 calls:
Slowest
23.0ms
(* (pow (/ 1 (pow re 2)) 1/3) (pow (cbrt -1) 2))
22.0ms
(* (pow (/ 1 (pow re 2)) 1/3) (pow (cbrt -1) 2))
14.0ms
(* (/ -1 re) (/ -1 re))
14.0ms
(* (/ -1 re) (/ -1 re))
7.0ms
(* (pow (/ 1 re) 1/3) (cbrt -1))

prune1.0s

Pruning

22 alts after pruning (20 fresh and 2 done)

Merged error: 5.6b

regimes824.0ms

Accuracy

54.4% (9.9b remaining)

Error of 17.4b against oracle of 7.5b and baseline of 29.2b

bsearch591.0ms

end0.0ms

sample35.7s

Algorithm
intervals
Results
28.0s7923×body10240exit
5.1s15613×body80nan
2.0s8001×body80valid