Average Error: 32.9 → 9.3
Time: 2.7m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1058513712164536 \cdot 10^{+149}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le -5.496700305081621 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\sqrt{(\left(a \cdot c\right) \cdot \left(-4\right) + \left(b \cdot b\right))_*} - b}}{a \cdot 2}\\ \mathbf{if}\;b \le 6.959770719239994 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target19.9
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1058513712164536e+149

    1. Initial program 62.5

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

    2. Taylor expanded around -inf 38.5

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

    3. Applied simplify1.5

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

    if -1.1058513712164536e+149 < b < -5.496700305081621e-102

    1. Initial program 42.0

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

    3. Applied flip--42.0

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

    4. Applied simplify13.7

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

    5. Applied simplify13.7

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

    if -5.496700305081621e-102 < b < 6.959770719239994e+151

    1. Initial program 11.5

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

    3. Applied clear-num11.6

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

    if 6.959770719239994e+151 < b

    1. Initial program 59.6

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

    2. Taylor expanded around inf 10.6

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\]

    3. Applied simplify2.3

      \[\leadsto \color{blue}{\frac{\frac{c}{1}}{b} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify9.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.1058513712164536 \cdot 10^{+149}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le -5.496700305081621 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot c\right) \cdot a}{\sqrt{(\left(a \cdot c\right) \cdot \left(-4\right) + \left(b \cdot b\right))_*} - b}}{a \cdot 2}\\ \mathbf{if}\;b \le 6.959770719239994 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018296 +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)))