Average Error: 19.2 → 13.1
Time: 58.5s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \left|\sqrt[3]{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{b - b}\\ \end{array} \le -6.802596353047467 \cdot 10^{+307}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \left|\sqrt[3]{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{b - b}\\ \end{array} \le 6.136807324467323 \cdot 10^{+223}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left|\sqrt[3]{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} - b}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if (if (>= b 0) (/ (- (- b) (* (fabs (cbrt (fma (- c) (* 4 a) (* b b)))) (sqrt (cbrt (fma (* a 4) (- c) (* b b)))))) (* 2 a)) (/ (* 2 c) (- b b))) < -6.802596353047467e+307 or 6.136807324467323e+223 < (if (>= b 0) (/ (- (- b) (* (fabs (cbrt (fma (- c) (* 4 a) (* b b)))) (sqrt (cbrt (fma (* a 4) (- c) (* b b)))))) (* 2 a)) (/ (* 2 c) (- b b)))

    1. Initial program 25.9

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

    2. Applied simplify25.9

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

    3. Taylor expanded around 0 17.4

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

    if -6.802596353047467e+307 < (if (>= b 0) (/ (- (- b) (* (fabs (cbrt (fma (- c) (* 4 a) (* b b)))) (sqrt (cbrt (fma (* a 4) (- c) (* b b)))))) (* 2 a)) (/ (* 2 c) (- b b))) < 6.136807324467323e+223

    1. Initial program 2.7

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

    2. Applied simplify2.7

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

    4. Applied add-cube-cbrt2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2} \cdot c}{\sqrt{\left(\sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}\right) \cdot \sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} - b}\\ \end{array}\]

    5. Applied sqrt-prod2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\sqrt{\sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} - b}\\ \end{array}\]

    6. Applied simplify2.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2} \cdot c}{\left|\sqrt[3]{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}\right| \cdot \sqrt{\sqrt[3]{(\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} - b}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 58.5s)Debug logProfile

herbie shell --seed 2018170 +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)))))))