Average Error: 35.3 → 31.6
Time: 35.6s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{1}{a \cdot 2}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{1}{a \cdot 2}}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  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. Using strategy rm

  3. Applied cbrt-prod33.3

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

  5. Applied cbrt-prod31.6

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

  6. Final simplification31.6

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

Reproduce

herbie shell --seed 2019053 +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))))))))