Average Error: 35.4 → 31.7
Time: 1.3m
Precision: 64
Internal Precision: 640
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le -1.3187864467551345 \cdot 10^{-82}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a + a} \cdot \frac{h \cdot h}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le 0.0:\\ \;\;\;\;\frac{\sqrt[3]{-\left(g + g\right)}}{\sqrt[3]{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}} + \frac{\sqrt[3]{\log \left(e^{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}\right)}}{\sqrt[3]{a + a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 3 regimes

  2. if (+ (cbrt (/ (+ (- g) g) (+ a a))) (cbrt (/ (- (- g) (sqrt (* (+ h g) (- g h)))) (+ a a)))) < -1.3187864467551345e-82

    1. Initial program 43.4

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

    3. Applied flip-+43.4

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

    4. Applied simplify43.1

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

    if -1.3187864467551345e-82 < (+ (cbrt (/ (+ (- g) g) (+ a a))) (cbrt (/ (- (- g) (sqrt (* (+ h g) (- g h)))) (+ a a)))) < 0.0

    1. Initial program 15.3

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

    3. Applied associate-*l/15.3

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

    4. Applied cbrt-div7.8

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

    5. Applied simplify7.8

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

    6. Taylor expanded around -inf 3.9

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

    7. Applied simplify3.9

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

    if 0.0 < (+ (cbrt (/ (+ (- g) g) (+ a a))) (cbrt (/ (- (- g) (sqrt (* (+ h g) (- g h)))) (+ a a))))

    1. Initial program 43.4

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

    3. Applied associate-*l/43.4

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

    4. Applied cbrt-div43.4

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

    5. Applied simplify43.4

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

    7. Applied associate-*l/43.4

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

    8. Applied cbrt-div41.8

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

    9. Applied simplify41.8

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

    11. Applied add-log-exp60.5

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

    12. Applied add-log-exp55.1

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

    13. Applied sum-log54.6

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

    14. Applied simplify42.5

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

  4. Applied simplify31.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le -1.3187864467551345 \cdot 10^{-82}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a + a} \cdot \frac{h \cdot h}{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a + a}}\\ \mathbf{if}\;\sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}} + \sqrt[3]{\frac{\left(-g\right) + g}{a + a}} \le 0.0:\\ \;\;\;\;\frac{\sqrt[3]{-\left(g + g\right)}}{\sqrt[3]{a + a}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}{a + a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a + a}} + \frac{\sqrt[3]{\log \left(e^{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}\right)}}{\sqrt[3]{a + a}}\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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))))))))