Average Error: 35.5 → 31.7
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\sqrt{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} \le 2.9815749658200832 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g} \cdot \sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (+ (* (cbrt (/ 1 (* 2 a))) (cbrt (* (sqrt (+ (- g) (sqrt (- (* g g) (* h h))))) (sqrt (+ (- g) (sqrt (- (* g g) (* h h)))))))) (/ (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (* 2 a)))) < 2.9815749658200832e+122

    1. Initial program 10.7

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

    3. Applied associate-*l/10.7

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

    4. Applied cbrt-div7.9

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

    5. Simplified7.9

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

    7. Applied associate-*l/7.9

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

    8. Applied cbrt-div4.9

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

    9. Simplified4.9

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

    11. Applied add-sqr-sqrt4.9

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

    if 2.9815749658200832e+122 < (+ (* (cbrt (/ 1 (* 2 a))) (cbrt (* (sqrt (+ (- g) (sqrt (- (* g g) (* h h))))) (sqrt (+ (- g) (sqrt (- (* g g) (* h h)))))))) (/ (cbrt (- (- g) (sqrt (* (+ g h) (- g h))))) (cbrt (* 2 a))))

    1. Initial program 61.5

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

    3. Applied associate-*l/61.5

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

    4. Applied cbrt-div60.6

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

    5. Simplified60.6

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

    6. Taylor expanded around inf 59.8

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

  4. Final simplification31.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\sqrt{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} \le 2.9815749658200832 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g} \cdot \sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018216 +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))))))))