Average Error: 33.7 → 6.5
Time: 39.3s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.0875233284862782 \cdot 10^{+96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.3855904099848335 \cdot 10^{-305}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{elif}\;b \le 8.566489273414817 \cdot 10^{+145}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.4
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.0875233284862782e+96

    1. Initial program 58.2

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Initial simplification58.2

      \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]

    3. Taylor expanded around 0 58.2

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    4. Simplified58.2

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]

    5. Taylor expanded around -inf 2.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    6. Simplified2.8

      \[\leadsto \color{blue}{\frac{-c}{b}}\]

    if -2.0875233284862782e+96 < b < -1.3855904099848335e-305

    1. Initial program 31.9

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Initial simplification31.9

      \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]

    3. Taylor expanded around 0 31.9

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    4. Simplified31.9

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv32.0

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{2 \cdot a}}\]
    7. Using strategy rm

    8. Applied flip--32.1

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}} \cdot \frac{1}{2 \cdot a}\]

    9. Applied associate-*l/32.2

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\]

    10. Simplified15.7

      \[\leadsto \frac{\color{blue}{\left(0 - \left(c \cdot -4\right) \cdot a\right) \cdot \frac{\frac{1}{2}}{a}}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]

    11. Taylor expanded around 0 8.0

      \[\leadsto \frac{\color{blue}{2 \cdot c}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]

    if -1.3855904099848335e-305 < b < 8.566489273414817e+145

    1. Initial program 9.1

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Initial simplification9.2

      \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]

    3. Taylor expanded around 0 9.1

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    4. Simplified9.2

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv9.3

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 8.566489273414817e+145 < b

    1. Initial program 57.5

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Initial simplification57.5

      \[\leadsto \frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{2 \cdot a}\]

    3. Taylor expanded around 0 57.5

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    4. Simplified57.5

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv57.5

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{2 \cdot a}}\]
    7. Using strategy rm

    8. Applied flip--62.3

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}} \cdot \frac{1}{2 \cdot a}\]

    9. Applied associate-*l/62.3

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) \cdot \left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} \cdot \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\]

    10. Simplified62.5

      \[\leadsto \frac{\color{blue}{\left(0 - \left(c \cdot -4\right) \cdot a\right) \cdot \frac{\frac{1}{2}}{a}}}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\]

    11. Taylor expanded around 0 2.3

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

    12. Simplified2.3

      \[\leadsto \color{blue}{\frac{-b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.0875233284862782 \cdot 10^{+96}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.3855904099848335 \cdot 10^{-305}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{elif}\;b \le 8.566489273414817 \cdot 10^{+145}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 39.3s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes28.76.55.323.494.9%
herbie shell --seed 2018340 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))