Average Error: 35.2 → 31.6
Time: 2.5m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.789869857912344 \cdot 10^{-204}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a + a}} \cdot \sqrt[3]{-\left(g + g\right)} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < 4.789869857912344e-204

    1. Initial program 35.7

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

    2. Applied simplify35.7

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

    4. Applied div-inv35.7

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

    5. Applied cbrt-prod32.0

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

    6. Taylor expanded around -inf 31.6

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

    7. Applied simplify31.6

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

    if 4.789869857912344e-204 < g

    1. Initial program 34.8

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

    2. Applied simplify34.8

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

    4. Applied cbrt-div31.6

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(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))))))))