Average Error: 35.1 → 31.0
Time: 3.4m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \frac{h \cdot h}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}} \le -1.9873215131330507 \cdot 10^{+182}:\\ \;\;\;\;\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\frac{h \cdot \left(-h\right)}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} + g}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{if}\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \frac{h \cdot h}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}} \le 4.110261436178516 \cdot 10^{+296}:\\ \;\;\;\;\frac{\sqrt[3]{h - \left(g + g\right)}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\frac{h \cdot \left(-h\right)}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} + g}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

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 (- (sqrt (* (- g h) (+ h g))) g)) (cbrt (* 2 a))) (cbrt (* (/ 1 (* 2 a)) (/ (* h h) (- (sqrt (* (- g h) (+ h g))) g))))) < -1.9873215131330507e+182 or 4.110261436178516e+296 < (+ (/ (cbrt (- (sqrt (* (- g h) (+ h g))) g)) (cbrt (* 2 a))) (cbrt (* (/ 1 (* 2 a)) (/ (* h h) (- (sqrt (* (- g h) (+ h g))) g)))))

    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. Using strategy rm

    3. Applied associate-*l/43.1

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

    4. Applied cbrt-div43.0

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

    5. Applied simplify43.0

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

    7. Applied cbrt-prod40.7

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

    9. Applied flip--40.8

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

    10. Applied simplify40.5

      \[\leadsto \frac{\sqrt[3]{\frac{\color{blue}{h \cdot \left(0 - h\right) + 0}}{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g}}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]

    if -1.9873215131330507e+182 < (+ (/ (cbrt (- (sqrt (* (- g h) (+ h g))) g)) (cbrt (* 2 a))) (cbrt (* (/ 1 (* 2 a)) (/ (* h h) (- (sqrt (* (- g h) (+ h g))) g))))) < 4.110261436178516e+296

    1. Initial program 17.6

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

    3. Applied associate-*l/17.6

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

    4. Applied cbrt-div11.6

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

    5. Applied simplify11.6

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

    6. Taylor expanded around -inf 10.0

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

    7. Applied simplify10.0

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

  4. Applied simplify31.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \frac{h \cdot h}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}} \le -1.9873215131330507 \cdot 10^{+182}:\\ \;\;\;\;\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\frac{h \cdot \left(-h\right)}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} + g}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{if}\;\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{a \cdot 2} \cdot \frac{h \cdot h}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g}} \le 4.110261436178516 \cdot 10^{+296}:\\ \;\;\;\;\frac{\sqrt[3]{h - \left(g + g\right)}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\frac{h \cdot \left(-h\right)}{\sqrt{\left(g - h\right) \cdot \left(g + h\right)} + g}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2020178 +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))))))))