Average Error: 35.0 → 31.5
Time: 2.1m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \left(\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{a \cdot 2}} \le -5.075969157073929 \cdot 10^{-112}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{if}\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \left(\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{a \cdot 2}} \le 7.216171240690485 \cdot 10^{-117}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{a}}{2}} \cdot \sqrt[3]{-\left(g + g\right)} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g} \cdot \sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}} - g}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a \cdot 2}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 3 regimes

  2. if (+ (cbrt (/ (- (sqrt (* (+ g h) (- g h))) g) (* a 2))) (cbrt (/ (- (- g) (* (* (cbrt (sqrt (* (+ g h) (- g h)))) (cbrt (sqrt (* (+ g h) (- g h))))) (cbrt (sqrt (* (+ g h) (- g h)))))) (* a 2)))) < -5.075969157073929e-112

    1. Initial program 9.2

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

    2. Applied simplify9.1

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

    4. Applied cbrt-div7.3

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

    if -5.075969157073929e-112 < (+ (cbrt (/ (- (sqrt (* (+ g h) (- g h))) g) (* a 2))) (cbrt (/ (- (- g) (* (* (cbrt (sqrt (* (+ g h) (- g h)))) (cbrt (sqrt (* (+ g h) (- g h))))) (cbrt (sqrt (* (+ g h) (- g h)))))) (* a 2)))) < 7.216171240690485e-117

    1. Initial program 61.2

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

    2. Applied simplify61.2

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

    4. Applied div-inv61.2

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

    5. Applied cbrt-prod43.0

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

    6. Taylor expanded around -inf 7.4

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

    7. Applied simplify7.2

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

    if 7.216171240690485e-117 < (+ (cbrt (/ (- (sqrt (* (+ g h) (- g h))) g) (* a 2))) (cbrt (/ (- (- g) (* (* (cbrt (sqrt (* (+ g h) (- g h)))) (cbrt (sqrt (* (+ g h) (- g h))))) (cbrt (sqrt (* (+ g h) (- g h)))))) (* a 2))))

    1. Initial program 43.1

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

    2. Applied simplify43.1

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

    4. Applied cbrt-div41.9

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

    6. Applied add-cube-cbrt41.9

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

    8. Applied add-cube-cbrt41.9

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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))))))))