Average Error: 33.3 → 8.0
Time: 39.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.360039612359337 \cdot 10^{+154}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5402771797889115 \cdot 10^{-292}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{elif}\;b \le 4.5644027550850554 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target21.1
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.360039612359337e+154

    1. Initial program 62.9

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

    2. Initial simplification62.9

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

    3. Taylor expanded around -inf 1.3

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

    4. Simplified1.3

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

    if -1.360039612359337e+154 < b < 2.5402771797889115e-292

    1. Initial program 33.2

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

    2. Initial simplification33.2

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

    4. Applied div-inv33.2

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

    6. Applied flip--33.3

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

    7. Applied associate-*l/33.4

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

    8. Simplified14.4

      \[\leadsto \frac{\color{blue}{\left(0 - \left(c \cdot a\right) \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}}}{\left(-b\right) + \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}\]

    9. Taylor expanded around -inf 8.7

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

    if 2.5402771797889115e-292 < b < 4.5644027550850554e-14

    1. Initial program 11.1

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

    2. Initial simplification11.1

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

    4. Applied div-inv11.2

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

    if 4.5644027550850554e-14 < b

    1. Initial program 29.9

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

    2. Initial simplification29.8

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

    3. Taylor expanded around inf 9.1

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.360039612359337 \cdot 10^{+154}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.5402771797889115 \cdot 10^{-292}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{elif}\;b \le 4.5644027550850554 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 39.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.88.05.923.891.4%
herbie shell --seed 2018351 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))