Average Error: 33.6 → 9.5
Time: 16.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2828914567326436 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.846288549570831 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.6
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2828914567326436e+154

    1. Initial program 60.9

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

    2. Simplified60.9

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

    4. Applied div-inv60.9

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

    5. Taylor expanded around -inf 2.0

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

    if -1.2828914567326436e+154 < b < 9.846288549570831e-88

    1. Initial program 11.5

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

    2. Simplified11.4

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

    4. Applied div-inv11.6

      \[\leadsto \color{blue}{\frac{\sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*} - b}{2} \cdot \frac{1}{a}}\]
    5. Using strategy rm

    6. Applied associate-*l/11.6

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

    7. Simplified11.4

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

    if 9.846288549570831e-88 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    4. Applied div-inv52.3

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

    5. Taylor expanded around inf 9.4

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

    6. Simplified9.4

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2828914567326436 \cdot 10^{+154}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.846288549570831 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))