Average Error: 35.2 → 30.6
Time: 2.4m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)} \le -1.738823845280941 \cdot 10^{-157}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-\left(g + g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{2 \cdot a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if (- (- g) (sqrt (* (- g h) (+ h g)))) < -1.738823845280941e-157

    1. Initial program 42.8

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

    2. Applied simplify42.8

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a \cdot 2}}}\]
    3. Using strategy rm

    4. Applied cbrt-div40.3

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

    if -1.738823845280941e-157 < (- (- g) (sqrt (* (- g h) (+ h g))))

    1. Initial program 18.5

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

    2. Applied simplify18.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a \cdot 2}}}\]
    3. Using strategy rm

    4. Applied cbrt-div11.6

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

    5. Taylor expanded around -inf 9.4

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

    6. Applied simplify9.4

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(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))))))))