Average Error: 33.2 → 10.0
Time: 2.2m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4259686058446102 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{if}\;b \le 1.056111401872344 \cdot 10^{-129}:\\ \;\;\;\;\frac{b + \left(-\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right)}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{\frac{2}{c}}}{\frac{c + c}{\frac{b}{a}} - \left(b + b\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

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

Derivation

  1. Split input into 3 regimes

  2. if b < -1.4259686058446102e+154

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 2.2

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

    3. Applied simplify2.2

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

    if -1.4259686058446102e+154 < b < 1.056111401872344e-129

    1. Initial program 10.4

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

    3. Applied frac-2neg10.4

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

    4. Applied simplify10.4

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

    if 1.056111401872344e-129 < b

    1. Initial program 50.6

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

    3. Applied flip-+50.6

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

    4. Applied simplify24.2

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

    5. Taylor expanded around inf 22.0

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

    6. Applied simplify11.4

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

    7. Applied simplify11.5

      \[\leadsto \frac{\color{blue}{\frac{4}{\frac{2}{c}}}}{\frac{c + c}{\frac{b}{a}} - \left(b + b\right)}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1063185673 2139736501 2393378123 1907444849 1070993796 1007244912)' 
(FPCore (a b c)
  :name "quadp (p42, positive)"

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

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