Average Error: 32.8 → 9.8
Time: 4.3m
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2596647331824642 \cdot 10^{-31}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.3977277462270081 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(-\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.8
Target20.2
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2596647331824642e-31

    1. Initial program 53.7

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

    2. Simplified53.7

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

    3. Taylor expanded around -inf 6.7

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

    4. Simplified6.7

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

    if -1.2596647331824642e-31 < b < 1.3977277462270081e+122

    1. Initial program 13.5

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

    2. Simplified13.5

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

    4. Applied *-un-lft-identity13.5

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

    5. Applied sqrt-prod13.5

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

    6. Applied add-cube-cbrt13.7

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

    7. Applied distribute-lft-neg-in13.7

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

    8. Applied prod-diff13.8

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

    9. Simplified13.6

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

    10. Simplified13.5

      \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right) + \color{blue}{0}}{2}}{a}\]
    11. Using strategy rm

    12. Applied add-sqr-sqrt13.5

      \[\leadsto \frac{\frac{\left(\left(-\sqrt{\color{blue}{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}}\right) - b\right) + 0}{2}}{a}\]

    13. Applied sqrt-prod13.7

      \[\leadsto \frac{\frac{\left(\left(-\color{blue}{\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}}\right) - b\right) + 0}{2}}{a}\]

    14. Applied distribute-rgt-neg-in13.7

      \[\leadsto \frac{\frac{\left(\color{blue}{\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \left(-\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right)} - b\right) + 0}{2}}{a}\]

    15. Applied fma-neg13.7

      \[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(-\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*} + 0}{2}}{a}\]

    if 1.3977277462270081e+122 < b

    1. Initial program 50.6

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

    2. Simplified50.6

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

    4. Applied *-un-lft-identity50.6

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

    5. Applied sqrt-prod50.6

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

    6. Applied add-cube-cbrt50.7

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

    7. Applied distribute-lft-neg-in50.7

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

    8. Applied prod-diff51.7

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

    9. Simplified51.6

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

    10. Simplified50.5

      \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - b\right) + \color{blue}{0}}{2}}{a}\]
    11. Using strategy rm

    12. Applied *-un-lft-identity50.5

      \[\leadsto \frac{\frac{\left(\left(-\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) - \color{blue}{1 \cdot b}\right) + 0}{2}}{a}\]

    13. Applied neg-mul-150.5

      \[\leadsto \frac{\frac{\left(\color{blue}{-1 \cdot \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}} - 1 \cdot b\right) + 0}{2}}{a}\]

    14. Applied prod-diff50.5

      \[\leadsto \frac{\frac{\color{blue}{\left((-1 \cdot \left(\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}\right) + \left(-b \cdot 1\right))_* + (\left(-b\right) \cdot 1 + \left(b \cdot 1\right))_*\right)} + 0}{2}}{a}\]

    15. Simplified50.6

      \[\leadsto \frac{\frac{\left(\color{blue}{\left(\left(-b\right) - \sqrt{(b \cdot b + \left(\left(-4 \cdot c\right) \cdot a\right))_*}\right)} + (\left(-b\right) \cdot 1 + \left(b \cdot 1\right))_*\right) + 0}{2}}{a}\]

    16. Simplified50.6

      \[\leadsto \frac{\frac{\left(\left(\left(-b\right) - \sqrt{(b \cdot b + \left(\left(-4 \cdot c\right) \cdot a\right))_*}\right) + \color{blue}{\left(\left(-b\right) + b\right)}\right) + 0}{2}}{a}\]

    17. Taylor expanded around inf 3.7

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2596647331824642 \cdot 10^{-31}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.3977277462270081 \cdot 10^{+122}:\\ \;\;\;\;\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(-\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))