Average Error: 34.5 → 30.8
Time: 1.5m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{1}{a + a}} + \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}}}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  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. Applied simplify34.5

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

  4. Applied div-inv34.5

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

  5. Applied cbrt-prod32.6

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

  7. Applied add-cube-cbrt32.7

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

  8. Applied add-cube-cbrt32.7

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

  9. Applied times-frac32.7

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

  10. Applied cbrt-prod30.8

    \[\leadsto \sqrt[3]{\left(-g\right) + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{1}{a + a}} + \color{blue}{\sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}}}}\]
  11. Removed slow pow expressions.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1063154770 1824007522 645063331 41291047 494775821 1237684644)' 
(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))))))))