Average Error: 35.6 → 32.1
Time: 2.5m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{h \cdot h}}{\sqrt[3]{\left(2 \cdot a\right) \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}} \le -3.20502593378069 \cdot 10^{+120}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \mathbf{if}\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{h \cdot h}}{\sqrt[3]{\left(2 \cdot a\right) \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}} \le 1.2278193223251025 \cdot 10^{+134}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{h \cdot h}}{\sqrt[3]{\left(2 \cdot a\right) \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-g\right) - \left(\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)}} \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)}}}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

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 (+ (/ (cbrt (- (sqrt (* (- g h) (+ g h))) g)) (cbrt (* 2 a))) (/ (cbrt (* h h)) (cbrt (* (* 2 a) (+ (- g) (sqrt (- (* g g) (* h h)))))))) < -3.20502593378069e+120

    1. Initial program 49.5

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

    3. Applied cbrt-prod47.6

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

    4. Taylor expanded around inf 47.8

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

    if -3.20502593378069e+120 < (+ (/ (cbrt (- (sqrt (* (- g h) (+ g h))) g)) (cbrt (* 2 a))) (/ (cbrt (* h h)) (cbrt (* (* 2 a) (+ (- g) (sqrt (- (* g g) (* h h)))))))) < 1.2278193223251025e+134

    1. Initial program 15.9

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

    3. Applied associate-*l/15.9

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

    4. Applied cbrt-div10.4

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

    5. Applied simplify10.4

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

    7. Applied flip--10.4

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

    8. Applied frac-times10.5

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

    9. Applied cbrt-div10.4

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

    10. Applied simplify9.5

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

    if 1.2278193223251025e+134 < (+ (/ (cbrt (- (sqrt (* (- g h) (+ g h))) g)) (cbrt (* 2 a))) (/ (cbrt (* h h)) (cbrt (* (* 2 a) (+ (- g) (sqrt (- (* g g) (* h h))))))))

    1. Initial program 39.9

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

    3. Applied associate-*l/39.9

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

    4. Applied cbrt-div36.7

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

    5. Applied simplify36.7

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

    7. Applied add-cube-cbrt37.1

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018201 +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))))))))