Average Error: 35.6 → 31.1
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -1.3552855110073527 \cdot 10^{-123}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \frac{h \cdot h}{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}}\\ \mathbf{if}\;g \le 2.863970870856417 \cdot 10^{-159}:\\ \;\;\;\;\sqrt[3]{h - \left(g + g\right)} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g} \cdot \sqrt{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]

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 3 regimes

  2. if g < -1.3552855110073527e-123

    1. Initial program 34.8

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

    3. Applied cbrt-prod31.4

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

    4. Applied simplify31.4

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

    6. Applied flip--31.4

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

    7. Applied simplify30.4

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

    8. Applied simplify30.4

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

    if -1.3552855110073527e-123 < g < 2.863970870856417e-159

    1. Initial program 46.7

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

    3. Applied cbrt-prod41.4

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

    4. Applied simplify41.4

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

    5. Taylor expanded around -inf 31.5

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

    6. Applied simplify31.5

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

    if 2.863970870856417e-159 < g

    1. Initial program 34.7

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

    3. Applied cbrt-prod34.6

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

    4. Applied simplify34.6

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

    6. Applied cbrt-prod30.9

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

    8. Applied add-sqr-sqrt31.6

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018195 
(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))))))))