2-ancestry mixing, positive discriminant

Average Error: 35.9 → 31.7

Time: 12.9s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;g \le -3.57937653452562965 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot 1}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{\frac{h \cdot h}{\frac{a}{1}}}{2}}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if g < -3.57937653452562965e-160

   1. Initial program 34.5

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

   2. Simplified 34.5

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

   4. Applied associate-*r/ 34.5

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

   5. Applied cbrt-div 31.0

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

   7. Applied flip-- 30.9

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

   8. Applied associate-*r/ 31.0

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

   9. Applied cbrt-div 31.0

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

   10. Simplified 31.0

       \[\leadsto \frac{\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot 1}}{\sqrt[3]{2 \cdot a}} + \frac{\color{blue}{\sqrt[3]{\frac{\frac{0 + h \cdot h}{\frac{a}{1}}}{2}}}}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}}\]

   11. Simplified 31.0

       \[\leadsto \frac{\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot 1}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{\frac{0 + h \cdot h}{\frac{a}{1}}}{2}}}{\color{blue}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}}\]

   if -3.57937653452562965e-160 < g

   1. Initial program 37.2

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

   2. Simplified 37.2

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

   4. Applied associate-*l/ 37.2

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

   5. Applied cbrt-div 33.5

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

   6. Taylor expanded around inf 32.4

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

4. Final simplification 31.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -3.57937653452562965 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot 1}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{\frac{h \cdot h}{\frac{a}{1}}}{2}}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (neg g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (neg g) (sqrt (- (* g g) (* h h))))))))