Average Error: 33.4 → 10.3
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.5063192545381745 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.201447365512736 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.6
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5063192545381745e+103

    1. Initial program 46.4

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

    2. Simplified46.4

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

    3. Taylor expanded around -inf 3.6

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

    if -1.5063192545381745e+103 < b < 1.201447365512736e-38

    1. Initial program 14.5

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

    2. Simplified14.5

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

    3. Taylor expanded around -inf 14.5

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

    4. Simplified14.5

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

    6. Applied associate-/l/14.5

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

    if 1.201447365512736e-38 < b

    1. Initial program 54.0

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

    2. Simplified54.0

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

    4. Applied div-inv54.0

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

    5. Taylor expanded around inf 7.3

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

    6. Simplified7.3

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5063192545381745 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.201447365512736 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 +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)))