Average Error: 34.5 → 30.7
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 2.111508374070247 \cdot 10^{-161}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{g \cdot g - \sqrt{g \cdot g - h \cdot h} \cdot \sqrt{g \cdot g - h \cdot h}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \end{array}\]

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 2 regimes

  2. if g < 2.111508374070247e-161

    1. Initial program 35.8

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

    3. Applied cbrt-prod32.4

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

    4. Simplified32.4

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

    5. Taylor expanded around -inf 31.9

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

    6. Simplified31.9

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

    if 2.111508374070247e-161 < g

    1. Initial program 33.0

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

    3. Applied cbrt-prod33.0

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

    4. Simplified33.0

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

    6. Applied associate-*l/33.0

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

    7. Applied cbrt-div29.3

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

    8. Simplified29.3

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

    10. Applied flip-+29.2

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

  4. Final simplification30.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 2.111508374070247 \cdot 10^{-161}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{g \cdot g - \sqrt{g \cdot g - h \cdot h} \cdot \sqrt{g \cdot g - h \cdot h}}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.630.729.33.459%
herbie shell --seed 2018354 +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))))))))