jeff quadratic root 1

Average Error: 20.4 → 6.9

Time: 6.1s

Precision: binary64

Math

\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.71075788809424871 \cdot 10^{87}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(\left(c \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot \frac{\sqrt[3]{a}}{b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(2 \cdot \frac{a}{b}\right) - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.3540792276628007 \cdot 10^{82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\left(-b\right) + \left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right)}} \cdot \frac{c}{\sqrt{\left(-b\right) + \left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right)}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.71075788809424871e87

   1. Initial program 30.1

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Taylor expanded around inf 30.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   3. Simplified 30.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \left(c \cdot \frac{a}{b}\right) - \left(b + b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   4. Taylor expanded around -inf 7.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}}\\ \end{array}\]

   5. Simplified 3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}}\\ \end{array}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{\color{blue}{1 \cdot b}}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}\\ \end{array}\]

   8. Applied add-cube-cbrt 3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{1 \cdot b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}\\ \end{array}\]

   9. Applied times-frac 3.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{1} \cdot \frac{\sqrt[3]{a}}{b}\right)}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}\\ \end{array}\]

   10. Applied associate-*r* 3.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \color{blue}{\left(\left(c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{1}\right) \cdot \frac{\sqrt[3]{a}}{b}\right)} - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}\\ \end{array}\]

   11. Simplified 3.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(\color{blue}{\left(c \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right)} \cdot \frac{\sqrt[3]{a}}{b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}\\ \end{array}\]

   if -1.71075788809424871e87 < b < 2.3540792276628007e82

   1. Initial program 9.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Simplified 9.3

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \end{array}}\]

   if 2.3540792276628007e82 < b

   1. Initial program 43.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Taylor expanded around inf 10.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   3. Simplified 4.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\color{blue}{2 \cdot \left(c \cdot \frac{a}{b}\right) - \left(b + b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   4. Taylor expanded around -inf 4.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{a \cdot c}{b} - b\right)}}\\ \end{array}\]

   5. Simplified 4.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(c \cdot \frac{a}{b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}}\\ \end{array}\]

   6. Taylor expanded around 0 4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}\\ \end{array}\]

   7. Simplified 4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}\\ \end{array}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 4.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\sqrt{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)} \cdot \sqrt{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}}}\\ \end{array}\]

   10. Applied times-frac 4.1

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}} \cdot \frac{c}{\sqrt{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}}\\ \end{array}\]

   11. Simplified 4.1

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\left(-b\right) + \left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right)}} \cdot \frac{c}{\sqrt{\left(-b\right) + \left(c \cdot \left(\frac{a}{b} \cdot 2\right) - b\right)}}\\ \end{array}\]

   12. Simplified 4.1

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2}{\sqrt{\left(-b\right) + \left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right)}} \cdot \frac{c}{\sqrt{\left(-b\right) + \left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right)}}}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.71075788809424871 \cdot 10^{87}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot \left(\left(c \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)\right) \cdot \frac{\sqrt[3]{a}}{b}\right) - \left(b + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(c \cdot \left(2 \cdot \frac{a}{b}\right) - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \le 2.3540792276628007 \cdot 10^{82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sqrt{\left(-b\right) + \left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right)}} \cdot \frac{c}{\sqrt{\left(-b\right) + \left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))