Average Error: 33.4 → 16.0
Time: 2.2m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 4.54571802761373 \cdot 10^{-290}:\\ \;\;\;\;\frac{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{if}\;b \le 5.345900402976386 \cdot 10^{+129}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{b + b}}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target21.0
Herbie16.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < 4.54571802761373e-290

    1. Initial program 21.1

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

    2. Applied simplify21.1

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

    4. Applied div-sub21.1

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

    if 4.54571802761373e-290 < b < 5.345900402976386e+129

    1. Initial program 34.3

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

    2. Applied simplify34.3

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

    4. Applied flip--34.4

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

    5. Applied simplify16.6

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

    7. Applied add-cube-cbrt17.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}}\]

    8. Applied simplify17.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right)} \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}\]

    9. Applied simplify9.7

      \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\]

    if 5.345900402976386e+129 < b

    1. Initial program 60.7

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

    2. Applied simplify60.8

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

    4. Applied flip--60.8

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

    5. Applied simplify36.1

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

    6. Taylor expanded around 0 14.1

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +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)))