Average Error: 33.2 → 9.1
Time: 58.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.325623142188407 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.649566014366079 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\ \mathbf{elif}\;b \le 5.853641604236724 \cdot 10^{+26}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot -2\right)}{\left(b + \sqrt{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.7
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.325623142188407e+151

    1. Initial program 59.5

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

    2. Simplified59.5

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

    3. Taylor expanded around -inf 2.1

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

    if -1.325623142188407e+151 < b < 1.649566014366079e-118

    1. Initial program 10.5

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

    2. Simplified10.5

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

    3. Taylor expanded around 0 10.5

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

    4. Simplified10.5

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

    if 1.649566014366079e-118 < b < 5.853641604236724e+26

    1. Initial program 39.0

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

    2. Simplified39.0

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

    4. Applied *-un-lft-identity39.0

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

    5. Applied div-inv39.0

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

    6. Applied times-frac39.0

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

    7. Simplified39.0

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

    8. Simplified39.0

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

    10. Applied flip--39.1

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

    11. Applied frac-times42.2

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

    12. Simplified21.9

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

    if 5.853641604236724e+26 < b

    1. Initial program 55.7

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

    2. Simplified55.7

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

    3. Taylor expanded around inf 5.0

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

    4. Simplified5.0

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.325623142188407 \cdot 10^{+151}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.649566014366079 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\ \mathbf{elif}\;b \le 5.853641604236724 \cdot 10^{+26}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot -2\right)}{\left(b + \sqrt{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(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)))