Average Error: 19.7 → 13.1
Time: 1.0m
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\frac{4 \cdot \left(c \cdot a\right)}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array} \le -7.566023914518227 \cdot 10^{+270}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{(\left(\frac{c}{b}\right) \cdot a + \left(-b\right))_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\frac{4 \cdot \left(c \cdot a\right)}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array} \le 2.46327387833595 \cdot 10^{+258}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b} \cdot \sqrt{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(\left(-c\right) \cdot \left(a \cdot 4\right) + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \end{array}\]

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) (/ (* c 2) (/ (* 4 (* c a)) (- (sqrt (fma (- c) (* 4 a) (* b b))) b))) (/ (- (sqrt (fma (- c) (* a 4) (* b b))) b) (* a 2))) < -7.566023914518227e+270

    1. Initial program 45.1

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

    2. Applied simplify45.0

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

    3. Taylor expanded around inf 27.7

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

    4. Applied simplify27.6

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

    if -7.566023914518227e+270 < (if (>= b 0) (/ (* c 2) (/ (* 4 (* c a)) (- (sqrt (fma (- c) (* 4 a) (* b b))) b))) (/ (- (sqrt (fma (- c) (* a 4) (* b b))) b) (* a 2))) < 2.46327387833595e+258

    1. Initial program 1.6

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

    2. Applied simplify1.6

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

    4. Applied add-sqr-sqrt1.8

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

    if 2.46327387833595e+258 < (if (>= b 0) (/ (* c 2) (/ (* 4 (* c a)) (- (sqrt (fma (- c) (* 4 a) (* b b))) b))) (/ (- (sqrt (fma (- c) (* a 4) (* b b))) b) (* a 2)))

    1. Initial program 41.6

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

    2. Applied simplify41.6

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

    3. Taylor expanded around 0 27.6

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))