Average Error: 30.4 → 18.1
Time: 7.0s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;-re \le -6.246958093959643 \cdot 10^{+128}:\\ \;\;\;\;\log re\\ \mathbf{if}\;-re \le -2.6954550274641055 \cdot 10^{-104}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;-re \le -8.601110130196432 \cdot 10^{-247}:\\ \;\;\;\;\log im\\ \mathbf{if}\;-re \le 1.3406729599668597 \cdot 10^{+82}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(-re\right)\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if (- re) < -6.246958093959643e+128

    1. Initial program 56.5

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

    2. Taylor expanded around inf 7.6

      \[\leadsto \log \color{blue}{re}\]

    if -6.246958093959643e+128 < (- re) < -2.6954550274641055e-104 or -8.601110130196432e-247 < (- re) < 1.3406729599668597e+82

    1. Initial program 19.6

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

    if -2.6954550274641055e-104 < (- re) < -8.601110130196432e-247

    1. Initial program 24.9

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

    2. Taylor expanded around 0 36.5

      \[\leadsto \log \color{blue}{im}\]

    if 1.3406729599668597e+82 < (- re)

    1. Initial program 47.4

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

    2. Taylor expanded around -inf 9.5

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

    3. Applied simplify9.5

      \[\leadsto \color{blue}{\log \left(-re\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 7.0s)Debug logProfile

herbie shell --seed 2018206 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))