Average Error: 35.3 → 35.5
Time: 55.9s
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\left(\frac{\sqrt[3]{\left(-g\right) + \left(-g\right)}}{2} \cdot \sqrt[3]{(-1 \cdot g + \left(\sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right))_*}\right) \cdot \frac{\sqrt[3]{(-1 \cdot g + \left((e^{\log_* (1 + \sqrt{\left(g + h\right) \cdot \left(g - h\right)})} - 1)^*\right))_*}}{a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\]

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 add-cube-cbrt35.4

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

  4. Taylor expanded around -inf 35.4

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

  5. Applied simplify35.4

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

  7. Applied expm1-log1p-u35.5

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

Runtime

Time bar (total: 55.9s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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))))))))