Average Error: 33.6 → 7.8
Time: 55.3s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.56619308291469 \cdot 10^{+141}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \le -1.9846299840027504 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.546772616322934 \cdot 10^{+41}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\ \end{array}\]

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

Original33.6
Target20.7
Herbie7.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.56619308291469e+141

    1. Initial program 56.6

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

    2. Initial simplification56.6

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

    3. Taylor expanded around -inf 2.2

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

    4. Simplified2.2

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

    if -7.56619308291469e+141 < b < -1.9846299840027504e-240

    1. Initial program 7.7

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

    2. Initial simplification7.7

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

    4. Applied add-sqr-sqrt7.7

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

    5. Applied sqrt-prod8.0

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

    if -1.9846299840027504e-240 < b < 4.546772616322934e+41

    1. Initial program 26.6

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

    2. Initial simplification26.6

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

    4. Applied flip--26.8

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

    5. Applied associate-/l/32.4

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

    6. Simplified23.4

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

    8. Applied associate-/r*17.1

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

    9. Taylor expanded around inf 11.0

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

    if 4.546772616322934e+41 < b

    1. Initial program 56.5

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

    2. Initial simplification56.5

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

    4. Applied flip--56.5

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

    5. Applied associate-/l/57.1

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

    6. Simplified27.8

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

    8. Applied associate-/r*26.0

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

    9. Taylor expanded around inf 24.5

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

    10. Taylor expanded around inf 6.7

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.56619308291469 \cdot 10^{+141}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \le -1.9846299840027504 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.546772616322934 \cdot 10^{+41}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot c}{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}\\ \end{array}\]

Runtime

Time bar (total: 55.3s)Debug logProfile

herbie shell --seed 2018250 
(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)))