Average Error: 35.5 → 30.9
Time: 1.8m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le -7.114524462453731 \cdot 10^{-107}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} + \left(-g\right)}{a + a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le 1.8695721932110265 \cdot 10^{-106}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a + a}} \cdot \sqrt[3]{\frac{\left(g + h\right) \cdot \left(g - h\right) - g \cdot g}{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - \left(-g\right)}} + \sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{1}{a + a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 3 regimes

  2. if (+ (cbrt (/ (+ g (- g)) (+ a a))) (cbrt (/ (- (- g) (sqrt (* (+ g h) (- g h)))) (+ a a)))) < -7.114524462453731e-107

    1. Initial program 43.6

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

    2. Applied simplify43.6

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

    4. Applied cbrt-div42.0

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

    if -7.114524462453731e-107 < (+ (cbrt (/ (+ g (- g)) (+ a a))) (cbrt (/ (- (- g) (sqrt (* (+ g h) (- g h)))) (+ a a)))) < 1.8695721932110265e-106

    1. Initial program 14.7

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

    2. Applied simplify14.7

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

    4. Applied div-inv14.7

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

    5. Applied cbrt-prod7.5

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

    6. Taylor expanded around -inf 2.6

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

    7. Applied simplify2.6

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

    if 1.8695721932110265e-106 < (+ (cbrt (/ (+ g (- g)) (+ a a))) (cbrt (/ (- (- g) (sqrt (* (+ g h) (- g h)))) (+ a a))))

    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

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

    4. Applied div-inv43.1

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

    5. Applied cbrt-prod43.1

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

    7. Applied div-inv43.1

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

    8. Applied cbrt-prod41.3

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

    10. Applied flip-+41.3

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

    11. Applied simplify41.3

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

  4. Applied simplify30.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le -7.114524462453731 \cdot 10^{-107}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} + \left(-g\right)}{a + a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le 1.8695721932110265 \cdot 10^{-106}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a + a}} \cdot \sqrt[3]{\frac{\left(g + h\right) \cdot \left(g - h\right) - g \cdot g}{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - \left(-g\right)}} + \sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\frac{1}{a + a}}\\ \end{array}}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' 
(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))))))))