Average Error: 37.3 → 5.9
Time: 26.8s
Precision: 64
Internal Precision: 3392
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{\sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot \left(\sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot \sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\right)} \le 3.76690728889376 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{1.0}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot 0.5\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original37.3
Target32.3
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* 0.5 (sqrt (* (* (cbrt (fma (hypot re im) 2.0 (* re 2.0))) (cbrt (fma (hypot re im) 2.0 (* re 2.0)))) (cbrt (fma (hypot re im) 2.0 (* re 2.0)))))) < 3.76690728889376e-310

    1. Initial program 59.9

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

    2. Initial simplification52.9

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

    4. Applied add-cube-cbrt52.9

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot \sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\right) \cdot \sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}}}\]

    5. Taylor expanded around -inf 44.2

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{e^{\left(\log \left(\frac{-1}{re}\right) + \log 1.0\right) - 2 \cdot \log \left(\frac{-1}{im}\right)}}}\]

    6. Simplified22.5

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{1.0}{\frac{-1}{im}}}}\]

    if 3.76690728889376e-310 < (* 0.5 (sqrt (* (* (cbrt (fma (hypot re im) 2.0 (* re 2.0))) (cbrt (fma (hypot re im) 2.0 (* re 2.0)))) (cbrt (fma (hypot re im) 2.0 (* re 2.0))))))

    1. Initial program 29.7

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

    2. Initial simplification0.3

      \[\leadsto 0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;0.5 \cdot \sqrt{\sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot \left(\sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot \sqrt[3]{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\right)} \le 3.76690728889376 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{1.0}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*} \cdot 0.5\\ \end{array}\]

Runtime

Time bar (total: 26.8s)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(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)))))