Average Error: 32.9 → 10.6
Time: 37.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3801296442484604 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.3540765153403165 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} + \left(-b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original32.9
Target20.6
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3801296442484604e+103

    1. Initial program 46.2

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

    3. Applied sub-neg46.2

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

    4. Taylor expanded around -inf 4.4

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

    if -1.3801296442484604e+103 < b < 4.3540765153403165e-10

    1. Initial program 15.1

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

    3. Applied sub-neg15.1

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

    if 4.3540765153403165e-10 < b

    1. Initial program 54.1

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

    2. Taylor expanded around inf 6.4

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

    3. Simplified6.4

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3801296442484604 \cdot 10^{+103}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.3540765153403165 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} + \left(-b\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 37.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.910.65.627.382%
herbie shell --seed 2018352 
(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)))