Average Error: 35.2 → 31.5
Time: 2.2m
Precision: 64
Internal Precision: 1408
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 3.3290768654776305 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}} + \sqrt[3]{\frac{\frac{g \cdot g - \left(g - h\right) \cdot \left(g + h\right)}{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < 3.3290768654776305e-140

    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

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

    4. Applied cbrt-div31.6

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

    6. Applied flip--32.0

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

    7. Applied simplify32.0

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

    if 3.3290768654776305e-140 < 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-div30.8

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

Runtime

Time bar (total: 2.2m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(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))))))))