Average Error: 35.3 → 31.8
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{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \frac{\sqrt[3]{\left(-g\right) + g}}{\sqrt[3]{a + a}} \le -1.785396009510573 \cdot 10^{-74}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{\sqrt{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt{a + a}}} + \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + \left(-g\right)}{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \frac{\sqrt[3]{\left(-g\right) + g}}{\sqrt[3]{a + a}} \le 4.9183340226481 \cdot 10^{-105}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{a + a}} + \frac{\sqrt[3]{-\left(g + g\right)}}{\sqrt[3]{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \frac{\sqrt[3]{\left(-g\right) + g}}{\sqrt[3]{a + a}} \le +\infty:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + \left(-g\right)}{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + \left(-g\right)}{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{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))) (cbrt (+ a a))) (* (cbrt (/ (* (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (- (- g) (sqrt (* (+ g h) (- g h)))))) (* (cbrt (+ a a)) (cbrt (+ a a))))) (cbrt (/ (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (+ a a)))))) < -1.785396009510573e-74

    1. Initial program 44.3

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

    2. Applied simplify44.3

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

    4. Applied add-sqr-sqrt45.1

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

    5. Applied *-un-lft-identity45.1

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

    6. Applied times-frac45.1

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

    7. Applied cbrt-prod43.3

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

    if -1.785396009510573e-74 < (+ (/ (cbrt (+ g (- g))) (cbrt (+ a a))) (* (cbrt (/ (* (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (- (- g) (sqrt (* (+ g h) (- g h)))))) (* (cbrt (+ a a)) (cbrt (+ a a))))) (cbrt (/ (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (+ a a)))))) < 4.9183340226481e-105

    1. Initial program 14.4

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

    2. Applied simplify14.4

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

    4. Applied cbrt-div8.3

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

    5. Taylor expanded around -inf 5.4

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

    6. Applied simplify5.4

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

    if 4.9183340226481e-105 < (+ (/ (cbrt (+ g (- g))) (cbrt (+ a a))) (* (cbrt (/ (* (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (- (- g) (sqrt (* (+ g h) (- g h)))))) (* (cbrt (+ a a)) (cbrt (+ a a))))) (cbrt (/ (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (+ a a))))))

    1. Initial program 43.0

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

    2. Applied simplify43.0

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

    4. Applied add-cube-cbrt43.1

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

    5. Applied *-un-lft-identity43.1

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

    6. Applied times-frac43.1

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

    7. Applied cbrt-prod41.4

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

  4. Applied simplify31.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \frac{\sqrt[3]{\left(-g\right) + g}}{\sqrt[3]{a + a}} \le -1.785396009510573 \cdot 10^{-74}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{\sqrt{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt{a + a}}} + \sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + \left(-g\right)}{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \frac{\sqrt[3]{\left(-g\right) + g}}{\sqrt[3]{a + a}} \le 4.9183340226481 \cdot 10^{-105}:\\ \;\;\;\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{a + a}} + \frac{\sqrt[3]{-\left(g + g\right)}}{\sqrt[3]{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}} + \frac{\sqrt[3]{\left(-g\right) + g}}{\sqrt[3]{a + a}} \le +\infty:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + \left(-g\right)}{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + \left(-g\right)}{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{\sqrt[3]{a + a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a + a} \cdot \sqrt[3]{a + a}}}\\ \end{array}}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))))))))