Average Error: 33.6 → 11.0
Time: 2.3m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -1.4883081608237057 \cdot 10^{+92}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;-b \le 7.191356327737354 \cdot 10^{-135}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{if}\;-b \le 2.787035193570525 \cdot 10^{-74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;-b \le 1.3326732363290293 \cdot 10^{-08}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.4
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -1.4883081608237057e+92

    1. Initial program 44.4

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

    2. Taylor expanded around inf 10.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]

    3. Applied simplify4.4

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

    if -1.4883081608237057e+92 < (- b) < 7.191356327737354e-135

    1. Initial program 10.5

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

    if 7.191356327737354e-135 < (- b) < 2.787035193570525e-74 or 1.3326732363290293e-08 < (- b)

    1. Initial program 51.5

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

    2. Taylor expanded around -inf 46.6

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

    3. Applied simplify10.1

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

    if 2.787035193570525e-74 < (- b) < 1.3326732363290293e-08

    1. Initial program 40.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify11.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -1.4883081608237057 \cdot 10^{+92}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;-b \le 7.191356327737354 \cdot 10^{-135}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{if}\;-b \le 2.787035193570525 \cdot 10^{-74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;-b \le 1.3326732363290293 \cdot 10^{-08}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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

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