Average Error: 33.3 → 10.1
Time: 44.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -501671907657.34106:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -5.460929706117908 \cdot 10^{-18}:\\ \;\;\;\;\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}\right)}\\ \mathbf{elif}\;b \le -3.71101226524406 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 5.213921241762901 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.3
Target20.8
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -501671907657.34106 or -5.460929706117908e-18 < b < -3.71101226524406e-66

    1. Initial program 52.8

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

    2. Initial simplification52.8

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

    3. Taylor expanded around inf 52.8

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

    4. Taylor expanded around -inf 8.3

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

    5. Simplified8.3

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

    if -501671907657.34106 < b < -5.460929706117908e-18

    1. Initial program 43.1

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

    2. Initial simplification43.1

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

    4. Applied flip--43.1

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

    5. Applied associate-/l/44.5

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

    6. Simplified15.6

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

    if -3.71101226524406e-66 < b < 5.213921241762901e+84

    1. Initial program 13.4

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

    2. Initial simplification13.4

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

    3. Taylor expanded around inf 13.4

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

    if 5.213921241762901e+84 < b

    1. Initial program 43.6

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

    2. Initial simplification43.6

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

    3. Taylor expanded around inf 43.6

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

    5. Applied *-un-lft-identity43.6

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

    6. Applied associate-/l*43.7

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

    8. Applied associate-/r/43.7

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

    9. Taylor expanded around 0 4.5

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

    10. Simplified4.5

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -501671907657.34106:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -5.460929706117908 \cdot 10^{-18}:\\ \;\;\;\;\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}\right)}\\ \mathbf{elif}\;b \le -3.71101226524406 \cdot 10^{-66}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 5.213921241762901 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Runtime

Time bar (total: 44.6s)Debug logProfile

herbie shell --seed 2018258 
(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)))