Average Error: 33.2 → 7.2
Time: 44.2s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.7437605092982 \cdot 10^{+133}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.213906532895171 \cdot 10^{-282}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(\frac{c}{\frac{-1}{4}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{2}}}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.7
Herbie7.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.7437605092982e+133

    1. Initial program 53.4

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

    2. Simplified53.4

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

    3. Taylor expanded around -inf 2.6

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

    if -5.7437605092982e+133 < b < 1.213906532895171e-282

    1. Initial program 8.3

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

    2. Simplified8.3

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

    4. Applied div-inv8.4

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

    if 1.213906532895171e-282 < b < 1.7760913692530284e+28

    1. Initial program 29.3

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

    2. Simplified29.3

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

    4. Applied div-inv29.3

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

    6. Applied flip--29.4

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

    7. Applied associate-*l/29.5

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

    8. Simplified17.4

      \[\leadsto \frac{\color{blue}{\frac{0 - \left(4 \cdot a\right) \cdot c}{\frac{a}{\frac{1}{2}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt18.1

      \[\leadsto \frac{\frac{0 - \left(4 \cdot a\right) \cdot c}{\frac{a}{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]

    11. Applied *-un-lft-identity18.1

      \[\leadsto \frac{\frac{0 - \left(4 \cdot a\right) \cdot c}{\frac{\color{blue}{1 \cdot a}}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]

    12. Applied times-frac18.0

      \[\leadsto \frac{\frac{0 - \left(4 \cdot a\right) \cdot c}{\color{blue}{\frac{1}{\sqrt{\frac{1}{2}}} \cdot \frac{a}{\sqrt{\frac{1}{2}}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]

    13. Applied *-un-lft-identity18.0

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 - \left(4 \cdot a\right) \cdot c\right)}}{\frac{1}{\sqrt{\frac{1}{2}}} \cdot \frac{a}{\sqrt{\frac{1}{2}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]

    14. Applied times-frac17.8

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{1}{\sqrt{\frac{1}{2}}}} \cdot \frac{0 - \left(4 \cdot a\right) \cdot c}{\frac{a}{\sqrt{\frac{1}{2}}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]

    15. Simplified17.8

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2}}} \cdot \frac{0 - \left(4 \cdot a\right) \cdot c}{\frac{a}{\sqrt{\frac{1}{2}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]

    16. Simplified10.9

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

    if 1.7760913692530284e+28 < 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

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

    4. Applied div-inv55.7

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

    5. Taylor expanded around inf 4.9

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

    6. Simplified4.9

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.7437605092982 \cdot 10^{+133}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.213906532895171 \cdot 10^{-282}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.7760913692530284 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(\frac{c}{\frac{-1}{4}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{1}{2}}}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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)))