Average Error: 35.1 → 34.1
Time: 1.1m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -1.6530885811875263 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot h}{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot h}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < -1.6530885811875263e-162

    1. Initial program 34.3

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

    3. Applied flip--34.2

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

    4. Applied simplify33.3

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

    if -1.6530885811875263e-162 < g

    1. Initial program 35.9

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

    3. Applied flip-+35.7

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

    4. Applied simplify34.8

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(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))))))))