Average Error: 35.2 → 31.1
Time: 1.6m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(-g\right) + \sqrt{\left(h + g\right) \cdot \left(g - h\right)} \le 1.4362987467290371 \cdot 10^{-241}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} + \left(-g\right)}{a + a}} + \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\left(-g\right) - g}{\sqrt[3]{a + a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) + \left(-g\right)}}{\sqrt[3]{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{a + a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if (+ (- g) (sqrt (* (+ h g) (- g h)))) < 1.4362987467290371e-241

    1. Initial program 14.5

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

    2. Applied simplify14.5

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

    4. Applied add-cube-cbrt14.7

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

    5. Applied *-un-lft-identity14.7

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

    6. Applied times-frac14.7

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

    7. Applied cbrt-prod8.0

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

    8. Taylor expanded around inf 7.2

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

    if 1.4362987467290371e-241 < (+ (- g) (sqrt (* (+ h g) (- g h))))

    1. Initial program 43.4

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

    2. Applied simplify43.4

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

    4. Applied cbrt-div40.9

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

    5. Taylor expanded around -inf 40.6

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

    6. Applied simplify40.6

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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))))))))