Average Error: 19.4 → 6.7
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} = -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} \le -5.664294735500338 \cdot 10^{-265}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} \le 1.2932842732724377 \cdot 10^{-245}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(\frac{c}{b}\right) \cdot a + \left(-b\right))_*}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} \le 4.477701981831227 \cdot 10^{+293}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(\frac{c}{b}\right) \cdot a + \left(-b\right))_*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))) < -inf.0

    1. Initial program 61.1

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

    2. Taylor expanded around inf 20.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

    3. Applied simplify16.0

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\frac{c}{1}}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}}\]

    if -inf.0 < (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))) < -5.664294735500338e-265 or 1.2932842732724377e-245 < (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))) < 4.477701981831227e+293

    1. Initial program 2.4

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

    4. Applied simplify2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\sqrt{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

    5. Applied simplify2.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\sqrt{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} - b}}}\\ \end{array}\]

    if -5.664294735500338e-265 < (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))) < 1.2932842732724377e-245 or 4.477701981831227e+293 < (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c))))))

    1. Initial program 40.1

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

    2. Taylor expanded around -inf 23.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}\\ \end{array}\]

    3. Applied simplify21.3

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(\frac{c}{b}\right) \cdot a + \left(-b\right))_*}\\ \end{array}}\]

    4. Taylor expanded around 0 11.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(\frac{c}{b}\right) \cdot a + \left(-b\right))_*}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify6.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} = -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} \le -5.664294735500338 \cdot 10^{-265}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} \le 1.2932842732724377 \cdot 10^{-245}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(\frac{c}{b}\right) \cdot a + \left(-b\right))_*}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}\\ \end{array} \le 4.477701981831227 \cdot 10^{+293}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{(\left(\frac{c}{b}\right) \cdot a + \left(-b\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))