Average Error: 30.1 → 0.2
Time: 54.1s
Precision: 64
Internal Precision: 4416
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re - im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re + im}} \cdot \left(\sqrt{\sqrt{re + im}} \cdot \sqrt{re - im}\right) + re\right)}\]

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. Initial program 30.1

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re - im \cdot im} + re\right)}\]
  2. Using strategy rm

  3. Applied difference-of-squares30.1

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\left(re + im\right) \cdot \left(re - im\right)}} + re\right)}\]

  4. Applied sqrt-prod0.1

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{re + im} \cdot \sqrt{re - im}} + re\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.1

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\sqrt{re + im} \cdot \sqrt{re + im}}} \cdot \sqrt{re - im} + re\right)}\]

  7. Applied sqrt-prod0.2

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{re + im}} \cdot \sqrt{\sqrt{re + im}}\right)} \cdot \sqrt{re - im} + re\right)}\]

  8. Applied associate-*l*0.2

    \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re + im}} \cdot \left(\sqrt{\sqrt{re + im}} \cdot \sqrt{re - im}\right)} + re\right)}\]

Runtime

Time bar (total: 54.1s)Debug logProfile

herbie shell --seed 2019053 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (+ (sqrt (- (* re re) (* im im))) re)))))