Average Error: 43.3 → 41.3
Time: 1.5m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\sqrt[3]{{b}^{6}} - \left(a \cdot c\right) \cdot 3} - b}{3 \cdot a} \le -8.863546427604501 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{{b}^{6}} - \left(a \cdot c\right) \cdot 3} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\left(\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_* \cdot (\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right) \cdot (e^{\log_* (1 + (\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*)} - 1)^*\right)}^{\frac{1}{3}}} - b}{3 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 2 regimes

  2. if (/ (- (sqrt (- (cbrt (pow b 6)) (* (* a c) 3))) b) (* 3 a)) < -8.863546427604501e-21

    1. Initial program 26.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Applied simplify26.8

      \[\leadsto \color{blue}{\frac{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{3 \cdot a}}\]
    3. Using strategy rm

    4. Applied add-cbrt-cube27.0

      \[\leadsto \frac{\sqrt{\color{blue}{\sqrt[3]{\left((\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_* \cdot (\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*\right) \cdot (\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}} - b}{3 \cdot a}\]

    5. Applied simplify27.0

      \[\leadsto \frac{\sqrt{\sqrt[3]{\color{blue}{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}}}} - b}{3 \cdot a}\]

    6. Taylor expanded around 0 30.3

      \[\leadsto \frac{\sqrt{\color{blue}{{\left({b}^{6}\right)}^{\frac{1}{3}} - 3 \cdot \left(c \cdot a\right)}} - b}{3 \cdot a}\]

    7. Applied simplify26.8

      \[\leadsto \color{blue}{\frac{\sqrt{\sqrt[3]{{b}^{6}} - \left(a \cdot c\right) \cdot 3} - b}{3 \cdot a}}\]

    if -8.863546427604501e-21 < (/ (- (sqrt (- (cbrt (pow b 6)) (* (* a c) 3))) b) (* 3 a))

    1. Initial program 61.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Applied simplify61.7

      \[\leadsto \color{blue}{\frac{\sqrt{(\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{3 \cdot a}}\]
    3. Using strategy rm

    4. Applied add-cbrt-cube61.7

      \[\leadsto \frac{\sqrt{\color{blue}{\sqrt[3]{\left((\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_* \cdot (\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*\right) \cdot (\left(3 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}} - b}{3 \cdot a}\]

    5. Applied simplify61.7

      \[\leadsto \frac{\sqrt{\sqrt[3]{\color{blue}{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}}}} - b}{3 \cdot a}\]
    6. Using strategy rm

    7. Applied pow1/357.3

      \[\leadsto \frac{\sqrt{\color{blue}{{\left({\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}\right)}^{\frac{1}{3}}}} - b}{3 \cdot a}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt57.3

      \[\leadsto \frac{\sqrt{{\left({\color{blue}{\left(\left(\sqrt[3]{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*}\right) \cdot \sqrt[3]{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*}\right)}}^{3}\right)}^{\frac{1}{3}}} - b}{3 \cdot a}\]

    10. Applied unpow-prod-down57.3

      \[\leadsto \frac{\sqrt{{\color{blue}{\left({\left(\sqrt[3]{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*}\right)}^{3} \cdot {\left(\sqrt[3]{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*}\right)}^{3}\right)}}^{\frac{1}{3}}} - b}{3 \cdot a}\]

    11. Applied simplify57.3

      \[\leadsto \frac{\sqrt{{\left(\color{blue}{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_* \cdot (\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)} \cdot {\left(\sqrt[3]{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*}\right)}^{3}\right)}^{\frac{1}{3}}} - b}{3 \cdot a}\]

    12. Applied simplify57.3

      \[\leadsto \frac{\sqrt{{\left(\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_* \cdot (\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right) \cdot \color{blue}{(\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*}\right)}^{\frac{1}{3}}} - b}{3 \cdot a}\]
    13. Using strategy rm

    14. Applied expm1-log1p-u57.3

      \[\leadsto \frac{\sqrt{{\left(\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_* \cdot (\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right) \cdot \color{blue}{(e^{\log_* (1 + (\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*)} - 1)^*}\right)}^{\frac{1}{3}}} - b}{3 \cdot a}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))