2-ancestry mixing, positive discriminant

Average Error: 36.3 → 32.3

Time: 21.7min

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;g \le 1.413726079038446 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot 1}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(-h \cdot h\right)}}{\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \left(2 \cdot a\right)}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if g < 1.413726079038446e-159

   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-*r/ 37.2

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

   5. Applied cbrt-div 33.6

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

   6. Taylor expanded around -inf 32.5

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

   7. Simplified 32.5

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

   if 1.413726079038446e-159 < g

   1. Initial program 35.3

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

   2. Simplified 35.3

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

   4. Applied cbrt-prod 31.5

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

   6. Applied flip-- 31.4

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

   7. Applied frac-times 32.6

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

   8. Applied cbrt-div 32.5

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

   9. Simplified 32.1

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

4. Final simplification 32.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 1.413726079038446 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot 1}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(-h \cdot h\right)}}{\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \left(2 \cdot a\right)}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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))))))))