Average Error: 33.8 → 10.4
Time: 2.1m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8467051498896273 \cdot 10^{+80}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \mathbf{if}\;b \le -6.8370322578253936 \cdot 10^{-105}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \mathbf{if}\;b \le 2.739372012628897 \cdot 10^{+83}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b} \cdot \sqrt[3]{-b}\right) \cdot \left(\sqrt[3]{-b}\right) + \left(-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right))_*}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{1}{a + a}\right) \cdot \left(\left(-b\right) - b\right) + \left(\frac{c}{b}\right))_*\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target21.0
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8467051498896273e+80

    1. Initial program 57.9

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

    2. Taylor expanded around -inf 14.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]

    3. Applied simplify2.9

      \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}\]

    if -1.8467051498896273e+80 < b < -6.8370322578253936e-105

    1. Initial program 41.5

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

    2. Taylor expanded around -inf 31.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]

    3. Applied simplify21.9

      \[\leadsto \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}\]

    if -6.8370322578253936e-105 < b < 2.739372012628897e+83

    1. Initial program 12.4

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

    3. Applied add-cube-cbrt12.6

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

    4. Applied fma-neg12.6

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

    if 2.739372012628897e+83 < b

    1. Initial program 42.3

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

    3. Applied clear-num42.3

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

    4. Taylor expanded around inf 11.3

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

    5. Applied simplify5.5

      \[\leadsto \color{blue}{(\left(\frac{1}{a + a}\right) \cdot \left(\left(-b\right) - b\right) + \left(\frac{c}{b}\right))_*}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +o rules:numerics
(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)))