Average Error: 33.5 → 6.3
Time: 2.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -6.749681311314221 \cdot 10^{+155}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;-b \le 1.454644076748988 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{if}\;-b \le 1.7067741582199526 \cdot 10^{+90}:\\ \;\;\;\;\frac{4 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{c}{b} \cdot \left(a \cdot 2\right) - \left(b + b\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.5
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -6.749681311314221e+155

    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 12.2

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

    3. Applied simplify1.9

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

    if -6.749681311314221e+155 < (- b) < 1.454644076748988e-268

    1. Initial program 8.9

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

    if 1.454644076748988e-268 < (- b) < 1.7067741582199526e+90

    1. Initial program 33.8

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

    3. Applied flip--33.8

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

    4. Applied simplify16.0

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

    5. Applied simplify16.0

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

    7. Applied *-un-lft-identity16.0

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

    8. Applied times-frac16.0

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

    9. Applied simplify8.0

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

    if 1.7067741582199526e+90 < (- b)

    1. Initial program 58.1

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

    3. Applied flip--58.1

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

    4. Applied simplify30.9

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

    5. Applied simplify31.0

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

    7. Applied *-un-lft-identity31.0

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

    8. Applied times-frac30.9

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

    9. Applied simplify28.6

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

    10. Taylor expanded around -inf 6.5

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

    11. Applied simplify2.7

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

  4. Applied simplify6.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -6.749681311314221 \cdot 10^{+155}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;-b \le 1.454644076748988 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{if}\;-b \le 1.7067741582199526 \cdot 10^{+90}:\\ \;\;\;\;\frac{4 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{c}{b} \cdot \left(a \cdot 2\right) - \left(b + b\right)}\\ \end{array}}\]

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))