Average Error: 43.9 → 41.8
Time: 2.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{\sqrt[3]{\sqrt{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}} \cdot \sqrt{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}}}} - b}{3 \cdot a} \le -3.04241201827595 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{\sqrt[3]{\sqrt{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}} \cdot \sqrt{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}}}} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\left({\left(\sqrt[3]{\sqrt{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}}} \cdot \sqrt[3]{\sqrt{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}}}\right)}^{3}\right)}^{\frac{1}{3}}} - b}{3 \cdot a}\\ \end{array}\]

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 (* (sqrt (pow (fma (- c) (* a 3) (* b b)) 3)) (sqrt (pow (fma (- c) (* a 3) (* b b)) 3))))) b) (* 3 a)) < -3.04241201827595e-22

    1. Initial program 27.7

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

    2. Applied simplify27.7

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

    4. Applied add-cbrt-cube27.9

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

    5. Applied simplify27.9

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

    7. Applied add-sqr-sqrt27.7

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

    if -3.04241201827595e-22 < (/ (- (sqrt (cbrt (* (sqrt (pow (fma (- c) (* a 3) (* b b)) 3)) (sqrt (pow (fma (- c) (* a 3) (* b b)) 3))))) b) (* 3 a))

    1. Initial program 61.8

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

    2. Applied simplify61.8

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

    4. Applied add-cbrt-cube61.8

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

    5. Applied simplify61.9

      \[\leadsto \frac{\sqrt{\sqrt[3]{\color{blue}{{\left((\left(-c\right) \cdot \left(a \cdot 3\right) + \left(b \cdot b\right))_*\right)}^{3}}}} - b}{3 \cdot a}\]
    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-cbrt-cube57.3

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

    10. Applied simplify57.3

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

    12. Applied add-sqr-sqrt57.3

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

    13. Applied cbrt-prod57.3

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +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)))