The quadratic formula (r2)

Average Error: 34.6 → 6.5

Time: 7.2min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.37934596949737259 \cdot 10^{155}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.21649659560100321 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot 4\right)\\ \mathbf{elif}\;b \le 5.50482210561115281 \cdot 10^{131}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.6
Target	21.2
Herbie	6.5

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

Derivation

1. Split input into 4 regimes

2. if b < -3.37934596949737259e155

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 1.8

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

   if -3.37934596949737259e155 < b < 4.21649659560100321e-259

   1. Initial program 33.6

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

   3. Applied flip-- 33.7

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

   4. Simplified 16.3

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

   5. Simplified 16.3

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

   7. Applied *-un-lft-identity 16.3

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

   8. Applied times-frac 16.3

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

   9. Simplified 8.6

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

   if 4.21649659560100321e-259 < b < 5.50482210561115281e131

   1. Initial program 8.4

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

   if 5.50482210561115281e131 < b

   1. Initial program 55.7

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

   3. Applied flip-- 63.8

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

   4. Simplified 62.7

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

   5. Simplified 62.7

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

   6. Taylor expanded around 0 2.5

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

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.37934596949737259 \cdot 10^{155}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.21649659560100321 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot 4\right)\\ \mathbf{elif}\;b \le 5.50482210561115281 \cdot 10^{131}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))