Average Error: 34.1 → 6.9
Time: 37.9s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.807794752624979 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -1.5019845142605294 \cdot 10^{-305}:\\ \;\;\;\;\left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.897841360812673 \cdot 10^{+130}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.1
Target21.0
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6.807794752624979e+136

    1. Initial program 55.1

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

    2. Taylor expanded around -inf 2.4

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

    if -6.807794752624979e+136 < b < -1.5019845142605294e-305

    1. Initial program 9.3

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

    2. Taylor expanded around -inf 9.3

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

    3. Simplified9.3

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

    5. Applied div-inv9.5

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

    if -1.5019845142605294e-305 < b < 2.897841360812673e+130

    1. Initial program 34.1

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

    2. Taylor expanded around -inf 34.1

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

    3. Simplified34.1

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

    5. Applied div-inv34.1

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

    7. Applied flip-+34.2

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

    8. Applied associate-*l/34.2

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

    9. Simplified15.1

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

    10. Taylor expanded around inf 8.9

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

    if 2.897841360812673e+130 < b

    1. Initial program 61.0

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

    2. Taylor expanded around inf 1.9

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

    3. Simplified1.9

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.807794752624979 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -1.5019845142605294 \cdot 10^{-305}:\\ \;\;\;\;\left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} + \left(-b\right)\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.897841360812673 \cdot 10^{+130}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 37.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.26.96.023.296%
herbie shell --seed 2018351 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))