Average Error: 52.5 → 51.6
Time: 43.0s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\sqrt{{\left((e^{\log_* (1 + (-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*)} - 1)^*\right)}^{\frac{1}{3}} \cdot \sqrt[3]{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_* \cdot (-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} - b}{a \cdot 3}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.5

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

  2. Initial simplification52.5

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

  4. Applied expm1-log1p-u52.5

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

  6. Applied add-cbrt-cube52.5

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

  8. Applied pow1/351.9

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

  10. Applied unpow-prod-down51.9

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

  11. Simplified51.6

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

  12. Final simplification51.6

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

Runtime

Time bar (total: 43.0s)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))