Average Error: 37.5 → 24.7
Time: 38.6s
Precision: 64
Internal Precision: 3456
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{\frac{1}{2} \cdot 2.0}{\frac{-1}{im}} \le -6.436265392040012 \cdot 10^{+268}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \mathbf{if}\;\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{\frac{1}{2} \cdot 2.0}{\frac{-1}{im}} \le -1.1633688800232677 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot {\left(e^{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\right)}^{\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}}\\ \mathbf{if}\;\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{\frac{1}{2} \cdot 2.0}{\frac{-1}{im}} \le 3.8759052411778 \cdot 10^{-313}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(re + re\right)}\\ \mathbf{if}\;\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{\frac{1}{2} \cdot 2.0}{\frac{-1}{im}} \le 2.557898934912673 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{\frac{1}{2} \cdot 2.0}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{if}\;\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{\frac{1}{2} \cdot 2.0}{\frac{-1}{im}} \le 9.996032980801339 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(im + re\right)}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original37.5
Target32.6
Herbie24.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if (* (/ (/ -1 re) (/ -1 im)) (/ (* 1/2 2.0) (/ -1 im))) < -6.436265392040012e+268 or 9.996032980801339e+154 < (* (/ (/ -1 re) (/ -1 im)) (/ (* 1/2 2.0) (/ -1 im)))

    1. Initial program 41.0

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

    2. Taylor expanded around 0 31.2

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

    if -6.436265392040012e+268 < (* (/ (/ -1 re) (/ -1 im)) (/ (* 1/2 2.0) (/ -1 im))) < -1.1633688800232677e-77

    1. Initial program 16.9

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

    3. Applied add-exp-log19.6

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{e^{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt21.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot e^{\color{blue}{\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}}\]

    6. Applied exp-prod21.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{{\left(e^{\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \cdot \sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\right)}^{\left(\sqrt[3]{\log \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\right)}}}\]

    if -1.1633688800232677e-77 < (* (/ (/ -1 re) (/ -1 im)) (/ (* 1/2 2.0) (/ -1 im))) < 3.8759052411778e-313

    1. Initial program 39.4

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

    2. Taylor expanded around inf 25.3

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

    if 3.8759052411778e-313 < (* (/ (/ -1 re) (/ -1 im)) (/ (* 1/2 2.0) (/ -1 im))) < 2.557898934912673e-124

    1. Initial program 61.3

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

    3. Applied add-exp-log61.3

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

    4. Taylor expanded around -inf 39.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{e^{\left(\log \frac{1}{2} + \log \left(\frac{-1}{re}\right)\right) - 2 \cdot \log \left(\frac{-1}{im}\right)}}}\]

    5. Applied simplify9.2

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{\frac{1}{2} \cdot 2.0}{\frac{-1}{im}}} \cdot 0.5}\]

    if 2.557898934912673e-124 < (* (/ (/ -1 re) (/ -1 im)) (/ (* 1/2 2.0) (/ -1 im))) < 9.996032980801339e+154

    1. Initial program 30.5

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

    3. Applied flip-+30.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Applied simplify18.2

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
  3. Recombined 5 regimes into one program.

Runtime

Time bar (total: 38.6s)Debug logProfile

herbie shell --seed '#(1070864556 424010669 783715395 1203517814 4070606583 4107618214)' 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))