Average Error: 33.5 → 8.9
Time: 1.2m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.0458247919990112 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot \left(-c\right)}{\frac{a \cdot c}{b} \cdot 2}\\ \mathbf{elif}\;b \le -6.4047998808879856 \cdot 10^{-288}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.0376815687842526 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot \left(-c\right)}{b + \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.7
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.0458247919990112e+156

    1. Initial program 60.9

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

    2. Initial simplification60.9

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

    4. Applied flip--62.4

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

    5. Applied associate-/l/62.4

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

    6. Simplified62.5

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

    8. Applied distribute-lft-neg-out62.5

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

    9. Applied distribute-frac-neg62.5

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

    10. Simplified62.4

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

    11. Taylor expanded around -inf 22.8

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

    if -3.0458247919990112e+156 < b < -6.4047998808879856e-288

    1. Initial program 8.6

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

    2. Initial simplification8.6

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

    4. Applied add-sqr-sqrt8.6

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

    5. Applied sqrt-prod8.8

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

    6. Applied fma-neg8.8

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

    if -6.4047998808879856e-288 < b < 1.0376815687842526e+151

    1. Initial program 33.8

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

    2. Initial simplification33.8

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

    4. Applied flip--33.9

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

    5. Applied associate-/l/37.9

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

    6. Simplified18.9

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

    8. Applied distribute-lft-neg-out18.9

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

    9. Applied distribute-frac-neg18.9

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

    10. Simplified8.3

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

    if 1.0376815687842526e+151 < b

    1. Initial program 62.6

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

    2. Initial simplification62.6

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

    4. Applied flip--62.6

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

    5. Applied associate-/l/62.6

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

    6. Simplified38.1

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

    8. Applied distribute-lft-neg-out38.1

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

    9. Applied distribute-frac-neg38.1

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

    10. Simplified37.8

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

    12. Applied clear-num37.8

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

    13. Taylor expanded around inf 2.3

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.0458247919990112 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot \left(-c\right)}{\frac{a \cdot c}{b} \cdot 2}\\ \mathbf{elif}\;b \le -6.4047998808879856 \cdot 10^{-288}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.0376815687842526 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot \left(-c\right)}{b + \sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c}}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))