Average Error: 35.6 → 31.6
Time: 2.6m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{h \cdot h}}{\sqrt[3]{\left(2 \cdot a\right) \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}} \le -1.5670095674560527 \cdot 10^{+146}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{h \cdot h}}{\sqrt[3]{\left(2 \cdot a\right) \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}} \le 8.327873752318306 \cdot 10^{+130}:\\ \;\;\;\;(\left(\sqrt[3]{\left(-g\right) - g}\right) \cdot \left(\sqrt[3]{\frac{1}{a \cdot 2}}\right) + \left(\sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{a \cdot 2}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if (+ (* (cbrt (/ 1 (* 2 a))) (cbrt (- (sqrt (- (* g g) (* h h))) g))) (/ (cbrt (* h h)) (cbrt (* (* 2 a) (+ (- g) (sqrt (- (* g g) (* h h)))))))) < -1.5670095674560527e+146 or 8.327873752318306e+130 < (+ (* (cbrt (/ 1 (* 2 a))) (cbrt (- (sqrt (- (* g g) (* h h))) g))) (/ (cbrt (* h h)) (cbrt (* (* 2 a) (+ (- g) (sqrt (- (* g g) (* h h))))))))

    1. Initial program 43.5

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

    3. Applied associate-*l/43.5

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

    4. Applied cbrt-div41.2

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

    5. Applied simplify41.2

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

    if -1.5670095674560527e+146 < (+ (* (cbrt (/ 1 (* 2 a))) (cbrt (- (sqrt (- (* g g) (* h h))) g))) (/ (cbrt (* h h)) (cbrt (* (* 2 a) (+ (- g) (sqrt (- (* g g) (* h h)))))))) < 8.327873752318306e+130

    1. Initial program 16.2

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

    3. Applied cbrt-prod9.8

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

    4. Applied simplify9.8

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

    5. Taylor expanded around -inf 8.1

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

    6. Applied simplify8.1

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018195 +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))))))))