Average Error: 35.5 → 31.6
Time: 1.3m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -2.3257175154861777 \cdot 10^{-189}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} + \left(-g\right)}{\sqrt[3]{a + a}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} + \left(-g\right)}{a + a}} + \sqrt[3]{\left(-g\right) - g} \cdot \sqrt[3]{\frac{1}{a + a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < -2.3257175154861777e-189

    1. Initial program 35.5

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

    2. Applied simplify35.5

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

    4. Applied add-cube-cbrt35.6

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

    5. Applied *-un-lft-identity35.6

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

    6. Applied times-frac35.6

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

    7. Applied cbrt-prod31.8

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

    if -2.3257175154861777e-189 < g

    1. Initial program 35.5

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

    2. Applied simplify35.5

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

    4. Applied div-inv35.5

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

    5. Applied cbrt-prod31.7

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

    6. Taylor expanded around inf 31.3

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(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))))))))