Average Error: 33.3 → 7.0
Time: 21.5s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.466382232385927 \cdot 10^{+88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.62563977430495 \cdot 10^{-243}:\\ \;\;\;\;\frac{\frac{c}{2} \cdot 4}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 4.697385256886561 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.466382232385927e+88

    1. Initial program 57.8

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

    2. Initial simplification57.9

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

    3. Taylor expanded around -inf 3.0

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

    4. Simplified3.0

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

    if -9.466382232385927e+88 < b < -1.62563977430495e-243

    1. Initial program 33.1

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

    2. Initial simplification33.1

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

    4. Applied flip--33.2

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

    5. Applied associate-/l/37.5

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

    6. Simplified20.0

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

    8. Applied times-frac15.0

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

    9. Simplified8.1

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

    10. Simplified8.1

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

    12. Applied associate-*r/7.9

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

    if -1.62563977430495e-243 < b < 4.697385256886561e+39

    1. Initial program 10.6

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

    2. Initial simplification10.6

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

    3. Taylor expanded around 0 10.6

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

    4. Simplified10.6

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

    if 4.697385256886561e+39 < b

    1. Initial program 34.9

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

    2. Initial simplification34.8

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

    3. Taylor expanded around inf 5.8

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.466382232385927 \cdot 10^{+88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.62563977430495 \cdot 10^{-243}:\\ \;\;\;\;\frac{\frac{c}{2} \cdot 4}{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}\\ \mathbf{elif}\;b \le 4.697385256886561 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 21.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.37.05.523.893.8%
herbie shell --seed 2018297 +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)))