Average Error: 33.3 → 10.2
Time: 30.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.0478552312406124 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.208200170069043 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\left(-b\right) - \sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.0478552312406124e-113

    1. Initial program 50.4

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

    2. Simplified50.4

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

    4. Applied div-inv50.4

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

    5. Taylor expanded around -inf 10.6

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

    6. Simplified10.6

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

    if -5.0478552312406124e-113 < b < 1.208200170069043e+98

    1. Initial program 11.8

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

    2. Simplified11.8

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

    4. Applied clear-num12.0

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

    if 1.208200170069043e+98 < b

    1. Initial program 44.9

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

    2. Simplified44.9

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

    4. Applied clear-num44.9

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

    5. Taylor expanded around 0 4.3

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

    6. Simplified4.3

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.0478552312406124 \cdot 10^{-113}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.208200170069043 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\left(-b\right) - \sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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)))