Average Error: 37.8 → 30.4
Time: 18.3s
Precision: 64
Internal Precision: 128
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.0330631390590382 \cdot 10^{+135}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \le -2.3164688983588397 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + im\right)} \cdot 0.5\\ \mathbf{elif}\;re \le -1.634784357398101 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.6158739146069824 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + im\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.2587878943940152 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original37.8
Target33.3
Herbie30.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if re < -1.0330631390590382e+135

    1. Initial program 61.8

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

    3. Applied add-sqr-sqrt62.1

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

    5. Applied add-sqr-sqrt62.2

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

    6. Applied associate-*r*62.1

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

    7. Taylor expanded around -inf 51.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{0}}\]

    if -1.0330631390590382e+135 < re < -2.3164688983588397e+39 or -1.634784357398101e-196 < re < 1.6158739146069824e-239

    1. Initial program 36.8

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

    3. Applied add-sqr-sqrt37.9

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

    4. Taylor expanded around 0 40.5

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

    if -2.3164688983588397e+39 < re < -1.634784357398101e-196 or 1.6158739146069824e-239 < re < 1.2587878943940152e+113

    1. Initial program 26.6

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

    if 1.2587878943940152e+113 < re

    1. Initial program 51.7

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

    3. Applied add-sqr-sqrt51.7

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

    4. Taylor expanded around inf 9.7

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification30.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.0330631390590382 \cdot 10^{+135}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \le -2.3164688983588397 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + im\right)} \cdot 0.5\\ \mathbf{elif}\;re \le -1.634784357398101 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.6158739146069824 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + im\right)} \cdot 0.5\\ \mathbf{elif}\;re \le 1.2587878943940152 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 
(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)))))