Average Error: 34.9 → 30.7
Time: 2.3m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 2.289557328467568 \cdot 10^{-178}:\\ \;\;\;\;\sqrt[3]{\frac{-\left(g + g\right)}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{1}{a + a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < 2.289557328467568e-178

    1. Initial program 36.3

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

    2. Applied simplify36.3

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt36.4

      \[\leadsto \sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{\color{blue}{\left(\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}\right) \cdot \sqrt[3]{a + a}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\]

    5. Applied *-un-lft-identity36.4

      \[\leadsto \sqrt[3]{\frac{\color{blue}{1 \cdot \left(\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right)}}{\left(\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}\right) \cdot \sqrt[3]{a + a}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\]

    6. Applied times-frac36.4

      \[\leadsto \sqrt[3]{\color{blue}{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}} \cdot \frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{\sqrt[3]{a + a}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\]

    7. Applied cbrt-prod32.7

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{\sqrt[3]{a + a}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\]

    8. Taylor expanded around -inf 31.9

      \[\leadsto \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\left(-g\right) + \color{blue}{-1 \cdot g}}{\sqrt[3]{a + a}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\]

    9. Applied simplify31.9

      \[\leadsto \color{blue}{\sqrt[3]{\frac{-\left(g + g\right)}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{a + a}}}\]

    if 2.289557328467568e-178 < g

    1. Initial program 33.4

      \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

    2. Applied simplify33.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}}\]
    3. Using strategy rm

    4. Applied div-inv33.4

      \[\leadsto \sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\color{blue}{\left(\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{1}{a + a}}}\]

    5. Applied cbrt-prod29.5

      \[\leadsto \sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \color{blue}{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{1}{a + a}}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))